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The seven-dimensional irreducible representation of the G2 root system has highest weight ${\omega }_1=2{\alpha }_1+{\alpha }_2$. Its crystal basis is the chain of weights $2{\alpha }_1+{\alpha }_2,{\alpha }_1+{\alpha }_2,{\alpha }_1,0,-{\alpha }_1,-{\alpha }_1-{\alpha }_2,-2{\alpha }_1-{\alpha }_2$, with successive edge colors $1,2,1,1,2,1$. Each step subtracts its color's simple root. The nonzero weights are the six short roots, each of multiplicity one; the zero weight also has multiplicity one.

For $V=L({\omega }_1)$ with the G2 root system numbered short-first,In the chain basis of the crystal of the seven-dimensional G2 representation, normalize $v_1=f_1{v}_0$, $v_2=f_2{v}_1$, $v_3=f_1{v}_2$. Then $v_0\wedge v_1$ and $v_0\wedge v_3-2v_1\wedge v_2$ are nonzero highest-weight vectors of weights ${\omega }_2$ and ${\omega }_1$; the identities $e_1{v}_1=v_0$, $e_1{v}_3=2v_2$, $e_2{v}_2=v_1$ verify both raising conditions. The Weyl complete reducibility theorem and G2 dimension polynomial exhaust the twenty-one dimensions of the exterior square. In the symmetric square, $v_0^2$ generates the twenty-seven-dimensional summand. Self-duality supplies a Lie-invariant bilinear form, which is symmetric because a nondegenerate alternating bilinear form cannot have odd dimension; its inverse gives the remaining invariant line.

For a short root $\alpha$ in the G2 root system, restriction of the Adjoint representation to the sl2 subalgebra associated with a root givesHere $V(n)$ is the irreducible sl2 Lie algebra representation of highest weight $n$ and dimension $n+1$. Every nonzero root vector $x\in {\mathfrak{g}}_{\alpha }$ has $\dim Z_{\mathfrak{g}}(x)=6$: the raising operator has a one-dimensional kernel in each of these six irreducible summands.

For the G2 root system with ${\alpha }_1$ short and ${\alpha }_2$ long, the fundamental weights are ${\omega }_1=2{\alpha }_1+{\alpha }_2$ and ${\omega }_2=3{\alpha }_1+2{\alpha }_2$. The positive coroots, relative to the simple coroots, have coordinates $(1,0),(0,1),(1,3),(2,3),(1,1),(1,2)$. The Weyl dimension formula therefore gives, for nonnegative integers $a,b$,The denominator is the product of the pairings of the Weyl vector with those coroots. The dimensions at ${\omega }_1,{\omega }_2,2{\omega }_1$ are seven, fourteen and twenty-seven.

The finite Dynkin diagram $G_2$ has two vertices joined by a triple bond, with the arrow toward the short root. The squared root-length ratio is three. It represents the G2 root system.

$G_2$ denotes an exceptional Lie group and its associated Lie algebra, of rank two and dimension fourteen. Its roots form the G2 root system. The compact real form is the automorphism group of the octonion algebra; other real forms and the complex Lie algebra have the same complex root-system type.

The longest element of a finite Weyl group is the element sending all positive roots of a root system to negative roots. It is the longest element of a finite Coxeter group. For the G2 root system it is $-I$: its twelve roots form two hexagons, and the dihedral reflection group of a root system contains the rotation through $\pi$. It determines the highest weight of a dual representation.

For a Cartan-Weyl basis, the Jacobi identity givesThus the bracket has joint Cartan weight $\alpha +\beta$. If $\beta =-\alpha$, it belongs to the zero-weight space, which is the Cartan subalgebra, and so is a linear combination ${\tilde{\alpha }}_i{H}_i$. If $\alpha +\beta$ is a nonzero root, its root space is one-dimensional and the bracket is proportional to $E_{\alpha +\beta }$. If that nonzero sum is not a root, the bracket vanishes. This proves the asserted forms without assuming every pair of roots has a root sum.

With the stated normalization, put $H_{\alpha }=2{\alpha }_i{H}_i/{\alpha }^2$. The commutators becomeThe adjoint relations make $J_3=H_{\alpha }/2$, $J_{+}=E_{\alpha }$, $J_{-}=E_{-\alpha }$ the usual SU(2) Lie algebra generators. Hence every finite-dimensional SU(2) irreducible component has $J_3$ eigenvalues $m=-j,-j+1,\dots ,j$, with $2j$ an integer. Therefore $H_{\alpha }$ has integer eigenvalues $2m$, including in the finite-dimensional Adjoint representation. Algebraically this is the sl2 subalgebra associated with a root.

If $[H_{\alpha },X_{\lambda }]=\lambda X_{\lambda }$, another use of the Jacobi identity givesThe bracket is therefore zero or lies in the eigenvalue-$\lambda +2$ space. Within one irreducible SU(2) ladder it is proportional to the next chosen vector $X_{\lambda +2}$. If the full eigenspace has multiplicity, the weight statement is the general one; proportionality to one named vector presupposes the ladder choice. Similarly $E_{-\alpha }$ lowers this eigenvalue by two, and the ladder weights run from $-n$ to $n$ in steps of two.

For a root vector $E_{\beta }$, its $H_{\alpha }$ weight isThe adjoint SU(2) ladder therefore provesFor nonparallel $\alpha ,\beta$, raising and lowering $E_{\beta }$ move its joint Cartan weight through the root string $\beta +k\alpha$. The finite irreducible ladder has weights symmetric about zero. Starting at $m_{\alpha \beta }$, reaching its partner $-m_{\alpha \beta }$ requires $k=-m_{\alpha \beta }$. The corresponding nonzero vector lies in the root space of $\beta -m_{\alpha \beta }\alpha$, proving the Weyl reflection property:There is no zero joint root in this string when $\alpha ,\beta$ are nonparallel. Its one-dimensional root spaces prevent two irreducible ladders with the same weight parity from overlapping, so the ladder argument applies to the entire string. For $\beta =\pm \alpha$, reflection simply interchanges the two opposite roots, which already exist by the adjoint relation.

For angle and length restrictions, set $p=2\alpha \cdot \beta /{\alpha }^2$ and $q=2\alpha \cdot \beta /{\beta }^2$. These Cartan integers have the same sign when nonzero and obeyFor nonparallel, nonorthogonal roots, $pq$ is an integer strictly between zero and four. The possibilities are consequently:

Orthogonal roots have $p=q=0$ and $\theta =\pi /2$. Parallel roots have $pq=4$; the integer pairs would allow only length multiples $1,2,1/2$. The stated reducedness condition excludes the latter two, so parallel roots are equal or opposite, with angles zero or $\pi$ and equal lengths. ThusNot every root system realizes every listed angle.

The orthogonal pair alone gives no ratio through $q/p$. To finish the length restriction for a simple Lie algebra, use irreducibility of its root system. Let $U$ span the Weyl group orbit of $\alpha$. Both $U$ and $U^{\perp }$ are invariant under the orthogonal root reflections. If a root had nonzero projections in both, reflection of its projection in $U$ would have a nonzero $U^{\perp }$ component, violating invariance. Hence every root is entirely in one of these subspaces. A proper nonzero $U$ would split the root system into two orthogonal parts, contradicting irreducibility. This proves that the Weyl orbit spans an irreducible root space.

In particular, some orbit root $\gamma$ is nonorthogonal to any chosen $\beta$, and ${\gamma }^2={\alpha }^2$. Applying the previous integer-pair restriction to $\gamma ,\beta$ therefore also handles orthogonal $\alpha ,\beta$:There can be at most two different root lengths: three lengths would have ratios two and three relative to the shortest, but then the longest-to-middle squared-length ratio would be $3/2$, which is forbidden. A reducible collection of orthogonal rank-one factors could have independently chosen scales; the simple-algebra assumption excludes that exception.

Choose a generic linear functional to define positive roots. The simple roots are those positive roots which cannot be written as sums of two positive roots; they form a basis in which all root coefficients are integers of one sign. Distinct simple roots have nonpositive inner product: if their inner product were positive, their root string would contain their difference, and whichever difference is positive would decompose one of them. In rank two, irreducibility excludes the orthogonal case. The allowed simple-root angles are therefore $120^{\circ },135^{\circ },150^{\circ }$, giving the three irreducible diagrams in the classification of rank-two root systems: A2 root system, B2 root system and G2 root system. Reflections in the two simple roots generate their respective six, eight and twelve roots.

Here are explicit simple-root choices and positive roots; adding their negatives gives each complete diagram.

The nonnegative coefficients in each positive-root list verify the chosen simple roots explicitly. The orthogonal four-root system $A_1\times A_1$ is rank two but reducible, so it is not a diagram for the simple algebra assumed here.

Define the Cartan matrix in the column-coroot convention:Using the ordered roots above givesThe alternative row-coroot convention transposes these matrices. Stating the convention and root ordering avoids a spurious disagreement in the unequal-length cases.

Use the course convention that a root system is finite and is a reduced root system satisfying the crystallographic root system condition. These hypotheses are essential for the angle restriction. For roots $r,s$, the two Cartan integerssatisfy $ab=4{\cos }^2\theta$. If $r,s$ are not parallel, the strict Cauchy-Schwarz inequality and integrality give $ab\in \{0,1,2,3\}$. These values give angles $\pi /2$, $\pi /3$ or $2\pi /3$, $\pi /4$ or $3\pi /4$, and $\pi /6$ or $5\pi /6$, respectively. If the roots are parallel, reducedness implies $s=\pm r$, giving $0$ or $\pi$. ThereforeHere are examples accounting for every angle. In the A1 root system, the pairs $(r,r)$ and $(r,-r)$ give $0,\pi$. Two roots from different summands of $A_1\oplus A_1$ give $\pi /2$. The six unit vectors at angles $k\pi /3$ form an A2 root system: taking one root at angle zero and another at $\pi /3$ or $2\pi /3$ gives those two angles. In the B2 root system $\{\pm e_1,\pm e_2,\pm e_1\pm e_2\}$, pair $e_1$ with $e_1+e_2$ or $-e_1+e_2$ to obtain $\pi /4,3\pi /4$. In a G2 root system, take short unit roots at angles $k\pi /3$ and long roots of length $\sqrt{3}$ at angles $\pi /6+k\pi /3$. A short root at zero paired with long roots at $\pi /6,5\pi /6$ gives the remaining angles.

Number the simple roots of the G2 root system so that ${\alpha }_1$ is short and ${\alpha }_2$ is long. In the Euclidean plane chooseTheir squared lengths are one and three, and their angle is $150$ degrees. The six positive roots areTogether with their negatives these form the twelve-root diagram: two regular hexagons of radii one and $\sqrt{3}$, rotated by $30$ degrees relative to one another. Solving the fundamental weight equations givesIn particular ${\omega }_1$ is the highest short root and ${\omega }_2$ the highest long root. The root diagram below marks both weights.

To evaluate the Weyl dimension formula, put $a=n_1$, $b=n_2$ and $\lambda =a{\omega }_1+b{\omega }_2$. The Weyl vector is $\rho ={\omega }_1+{\omega }_2$. In the order of positive roots listed above, their coroots have simple-coroot coefficient pairsFor example the short root ${\alpha }_1+{\alpha }_2$ has coroot ${\alpha }_1^{\vee }+3{\alpha }_2^{\vee }$, while the long root $3{\alpha }_1+2{\alpha }_2$ has coroot ${\alpha }_1^{\vee }+2{\alpha }_2^{\vee }$. Pairing with $\lambda +\rho$ and dividing by the same factors at $a=b=0$ gives the G2 dimension polynomialThe denominator is $1\cdot 1\cdot 2\cdot 3\cdot 4\cdot 5$. As checks, the fundamental representations have dimensions seven and fourteen, and $a=b=0$ gives the trivial representation of dimension one.

Number the simple roots so that ${\alpha }_1$ is short and ${\alpha }_2$ is long. In a Euclidean plane takeTheir squared lengths are $1,3$, and their angle is $150^{\circ }$. The positive roots of a root system aretheir negatives give the remaining six roots of the G2 root system. The three short positive roots of a root system are ${\alpha }_1$, ${\alpha }_1+{\alpha }_2$, and $2{\alpha }_1+{\alpha }_2$.

Solving $⟨{\omega }_j,{\alpha }_i^{\vee }⟩={\delta }_{ij}$ gives the fundamental weightsIn this case both fundamental weights are themselves roots; the diagram marks their locations on the two hexagons.

Let $a=n_1$ and $b=n_2$. Relative to the simple coroots, the six positive coroots, in the root order above, have coordinatesFor example, ${\alpha }_1+{\alpha }_2$ has squared length one, so its coroot is $2{\alpha }_1+2{\alpha }_2={\alpha }_1^{\vee }+3{\alpha }_2^{\vee }$; the analogous length computation gives each other entry. Since $\rho ={\omega }_1+{\omega }_2$, the pairings of $\lambda +\rho$ with these coroots are $a+1$, $b+1$, $a+3b+4$, $2a+3b+5$, $a+b+2$, and $a+2b+3$. Their values at $a=b=0$ multiply to $1\cdot 1\cdot 4\cdot 5\cdot 2\cdot 3=120$. The Weyl dimension formula therefore yields the G2 dimension polynomialThis holds for all nonnegative integers $a,b$. In particular it gives dimensions $7$ and $14$ for $L({\omega }_1)$ and $L({\omega }_2)$, and $27$ for $L(2{\omega }_1)$. Reversing the numbering of the simple roots would interchange $a,b$ in the polynomial, so the short-first convention matters.

Under the Principal sl2 subalgebra, a root space of root $\beta$ has $h$-eigenvalue $\beta (2{\rho }^{\vee })=2\mathrm{ht}(\beta )$, where the height is the sum of its simple-root coefficients. The six positive roots of a root system of the G2 root system have heightsEach positive and negative root space has dimension one. The two-dimensional Cartan subalgebra contributes two copies of the zero weight. Therefore the Principal q-character of the G2 adjoint representation isEquivalently,The classification of finite-dimensional sl2 representations therefore gives the actual restricted-module decomposition $\mathfrak{g}\cong L_{10}\oplus L_2$, of dimensions $11+3=14$. In the convention using $q^{h/2}$, the same result is $q^5+q^4+q^3+q^2+2q+2+2q^{-1}+q^{-2}+q^{-3}+q^{-4}+q^{-5}$. Both forms encode the same two weight strings.

Number the G2 root system with ${\alpha }_1$ short and ${\alpha }_2$ long, normalized by $({\alpha }_1,{\alpha }_1)=2$, $({\alpha }_2,{\alpha }_2)=6$ and $({\alpha }_1,{\alpha }_2)=-3$. Its positive roots are ${\alpha }_1,{\alpha }_2,{\alpha }_1+{\alpha }_2,2{\alpha }_1+{\alpha }_2,3{\alpha }_1+{\alpha }_2,3{\alpha }_1+2{\alpha }_2$, and the negatives complete the twelve-root system. The short roots form one hexagon, and the long roots form a rotated hexagon whose radius is $\sqrt{3}$ times larger. The picture also supplies the crystal requested in the next part.

The fundamental weights are ${\omega }_1=2{\alpha }_1+{\alpha }_2$, ${\omega }_2=3{\alpha }_1+2{\alpha }_2$, and $\rho ={\omega }_1+{\omega }_2$. The six positive coroots, in coordinates relative to ${\alpha }_1^{\vee },{\alpha }_2^{\vee }$, are $(1,0),(0,1),(1,3),(2,3),(1,1),(1,2)$. Applying the Weyl dimension formula to $\lambda =n_1{\omega }_1+n_2{\omega }_2$ therefore gives the G2 dimension polynomialThe denominator is the product $1\cdot 1\cdot 2\cdot 3\cdot 4\cdot 5$ of the corresponding pairings with $\rho$. As checks, the weights ${\omega }_1,{\omega }_2,2{\omega }_1$ give dimensions $7,14,27$ respectively.

The triple bond in the original Dynkin diagram gives the Cartan integerswhere $\alpha$ is the long root and $\beta$ the short root. Their product is $4{\cos }^2\theta =3$, and the inner product of distinct simple roots is nonpositive. The ratio of the two integers gives the squared-length ratio. ThereforeIt is convenient to normalize $(\alpha ,\alpha )=6$, $(\beta ,\beta )=2$ and $(\alpha ,\beta )=-3$. Scaling the inner product does not affect the root system or the fundamental-weight relations.

For nonproportional roots $\gamma ,\delta$, the root-string theorem states that the root string is consecutive:The endpoints are maximal, and reflection in $\gamma$ reverses the string. The Cartan integer determines $p-q$, not in general the total $p+q+1$ by itself. This result follows by restricting the Adjoint representation to the sl2 subalgebra associated with a root. If the roots are distinct simple roots, $\delta -\gamma$ cannot be a root: its simple-root coefficients have opposite signs. Thus $p=0$ andHere length means number of roots; the number of intervals between them is one less. Distinctness matters. If $\gamma =\delta$, a reduced root system gives the set $\{\gamma ,-\gamma \}$, with a missing zero between them, so the consecutive-string theorem and the displayed simple-root formula do not apply.

In the G2 root system, the initial strings areFor $\delta =\alpha +3\beta$, $⟨\delta ,{\alpha }^{\vee }⟩=-1$. Since $\delta -\alpha =3\beta$ is not a root in a reduced root system, $p=0$ and $q=1$: this generates $2\alpha +3\beta$. The remaining strings explain why the construction stops. The $\alpha$-strings through $\beta$ and $\alpha +\beta$ are the same two-element string; the one through $\alpha +2\beta$ is a singleton since subtracting $\alpha$ gives $2\beta$, and its Cartan integer is zero; the one through $2\alpha +3\beta$ is the string $\{\alpha +3\beta ,2\alpha +3\beta \}$. The $\beta$-strings through $\alpha$, $\alpha +\beta$, $\alpha +2\beta$ and $\alpha +3\beta$ are the initial four-element string. Finally, $2\alpha +3\beta$ is orthogonal to $\beta$; its $\beta$-string is a singleton because $2\alpha +2\beta =2(\alpha +\beta )$ is not a root. Apply the same reasoning to negatives. Using the permitted completeness of this procedure givesThe short positive roots are $\beta ,\alpha +\beta ,\alpha +2\beta$, of squared length two; the other three are long, of squared length six. Each root space is one-dimensional and the Cartan subalgebra has dimension two, soHere the dimension refers to the Lie algebra, with one Cartan generator per rank, not just the number of roots.

Write a prospective weight as $w=a\alpha +b\beta$. The pairings with simple coroots areThe fundamental weights are dual to those coroots. Solving the two linear systems gives, in the long-root-first numbering of this paper,The representation with Dynkin labels $(0,1)$ has highest weight $\lambda ={\omega }_2$, a short root. Numbering the short root first, as some references do, would call this the $(1,0)$ representation instead; the representation itself is unchanged.

The weight set of a finite-dimensional irreducible highest-weight representation is invariant under the Weyl group and lies in the convex hull of the orbit of its highest weight. All weights also differ from the highest weight by an element of the root lattice. The orbit of $\lambda$ comprises the six short roots, so all six are weights. The lowering operators give the chainIn particular zero occurs: at weight $\beta$, its pairing with ${\beta }^{\vee }$ is two, so the lowering operator is nonzero by the finite-dimensional sl2 Lie algebra representation theory. The $\beta$-string through it is the usual three-weight string $\beta ,0,-\beta$; higher weight $2\beta$ would lie outside the highest-weight convex hull.

There can be no further weights. Every point of that convex hull has squared norm at most $\Vert \lambda {\Vert }^2=2$, while a root-lattice point hasThe integer solutions of the bound $\Vert w{\Vert }^2\le 2$ are precisely zero and the six short roots: $a=0$ gives $b=-1,0,1$; $a=1$ gives $b=1,2$; and $a=-1$ gives $b=-1,-2$. ThusThe final dimension uses the stipulated nondegeneracy of the weights. It also agrees with the Weyl dimension formula without that stipulation. The zero weight is not in the Weyl orbit of the nonzero weights, so this is not a minuscule representation.

In the diagram above, a coordinate label $(a,b)$ means $a\alpha +b\beta$. The left panel contains all twelve roots of a root system; the right panel contains the six short-root weights and the zero weight. The long-root-first convention is the same as in the calculations.

Choose the given short root $\alpha$ as a simple root, and let $\beta$ be the long simple root. This entails no loss of generality: the Weyl group is transitive on the short roots of the G2 root system. The relevant Cartan integers are $⟨\beta ,{\alpha }^{\vee }⟩=-3$ and $⟨\alpha ,{\beta }^{\vee }⟩=-1$. The positive roots areUnder ${\mathfrak{m}}_{\alpha }\cong {\mathfrak{sl}}_2$, the root space for $m\alpha +n\beta$ has $h_{\alpha }$ eigenvalue $2m-3n$. The subalgebra ${\mathfrak{m}}_{\alpha }$ itself is the three-dimensional irreducible $V(2)$ in the Adjoint representation.

The four root spaces along the root string $\beta ,\alpha +\beta ,2\alpha +\beta ,3\alpha +\beta$ have eigenvalues $-3,-1,1,3$. Consecutive spaces are connected by nonzero raising operators and lowering operators, so their sum is $V(3)$. The negative root string supplies another $V(3)$. The root spaces for $\pm (3\alpha +2\beta )$ have eigenvalue zero, and neither adding nor subtracting $\alpha$ gives a root; they are two copies of $V(0)$. Finally, the one-dimensional space $\ker \alpha \subset \mathfrak{t}$ commutes with ${\mathfrak{m}}_{\alpha }$, giving one more $V(0)$. We have accounted for all $14$ dimensions, and therefore the G2 adjoint branching to a short-root sl2 subalgebra isFor a nonzero root vector $x\in {\mathfrak{g}}_{\alpha }$, its Adjoint representation action is a nonzero scalar multiple of the raising operator on each summand. On each irreducible $V(n)$ this operator has a one-dimensional kernel, including $V(0)$. Since the Lie algebra centralizer is that kernel,

Use the wiki's Cartan matrix conventionLabel the long simple root first when the lengths differ. The matrix of an irreducible rank-two root system has the form $A=\left(\begin{array}{cc}2& -a\\ -b& 2\end{array}\right)$, where $a,b$ are positive integers. Integrality and the signs follow from the Cartan integers for distinct simple roots, and irreducibility rules out a zero off-diagonal pair. Its positive-definite symmetrization gives $ab<4$, equivalently $ab=4{\cos }^2\theta <4$. Moreover $a/b=\Vert {\alpha }_1{\Vert }^2/\Vert {\alpha }_2{\Vert }^2\ge 1$. Thus $(a,b)=(1,1),(2,1),(3,1)$, giving the entire classification of rank-two root systems relevant to a simple algebra:These correspond to ${\mathfrak{sl}}_3(\mathbb{C})$, ${\mathfrak{so}}_5(\mathbb{C})\cong {\mathfrak{sp}}_4(\mathbb{C})$, and the exceptional algebra ${\mathfrak{g}}_2$. Here $B_2=C_2$ denotes the Isomorphism between so5 and sp4, not an additional case. The disconnected $A_1\times A_1$ system corresponds to a semisimple but nonsimple algebra and is excluded.

The Dynkin diagrams, with nodes in the same order as the matrices, areThe second and third diagrams have two and three bonds respectively, with the arrowhead pointing to the short root ${\alpha }_2$. In the last diagram $\equiv$ draws the three bonds and $>$ their arrowhead.

To enumerate roots, use the standard root string theorem: for $\beta \ne \pm \alpha$, the roots $\beta +n\alpha$ are consecutive from $\beta -p\alpha$ through $\beta +q\alpha$, with $p-q=⟨\beta ,{\alpha }^{\vee }⟩$. Also use the standard results that the root system is reduced, every root is conjugate under the Weyl group to a simple root, and each root space has dimension one.

Write $m=a\in \{1,2,3\}$. Since a difference of simple roots is not a root, the ${\alpha }_2$-string starting at ${\alpha }_1$ has $p=0$ and $q=m$. It produces ${\alpha }_1,{\alpha }_1+{\alpha }_2,\dots ,{\alpha }_1+m{\alpha }_2$. The ${\alpha }_1$-string starting at ${\alpha }_2$ has $q=1$. In type $G_2$, the further string through $\beta ={\alpha }_1+3{\alpha }_2$ hasand $\beta -{\alpha }_1=3{\alpha }_2$ is not a root, so it also produces $2{\alpha }_1+3{\alpha }_2$. The resulting positive roots areAll roots are these and their negatives. To check completeness, the simple root reflections act on coordinates byEach listed set together with its negatives is stable under both reflections. It contains the simple roots and consists of roots already forced by strings. Since every root lies in a Weyl orbit of a simple root, no other roots can occur. This proves the lists for the A2 root system, B2 root system, and G2 root system. The root-space decomposition then gives

For the restriction to a sl2 subalgebra associated with a root, normalize its Cartan element to be $h_{\alpha }={\alpha }^{\vee }$. On a root vector of root $\beta$, its eigenvalue is $⟨\beta ,{\alpha }^{\vee }⟩$. The classification of finite-dimensional sl2 representations says that $R(\Lambda )$ has weights $\Lambda ,\Lambda -2,\dots ,-\Lambda$ and dimension $\Lambda +1$. A root string of $\Lambda +1$ nonparallel roots therefore supplies one such irreducible module: the raising and lowering brackets connect its consecutive root spaces.

For clarity, the positive-side strings for each simple-root direction are listed below. Brackets denote the whole consecutive string; a single listed root is a string of length one. In every row also include each negative string in reverse order, and separately the triple formed by the root ${\alpha }_i$, its negative, and $h_{{\alpha }_i}$.The root's own triple is always $R(2)$. The one-dimensional subspace of $\mathfrak{h}$ annihilated by ${\alpha }_i$ commutes with this subalgebra and gives $R(0)$. Each singleton root in the table, and its negative, contributes another $R(0)$. Reading the other string lengths now gives the adjoint branching to root sl2 subalgebras in rank two:Their dimensions are respectively $8,10,10,14,14$. The last row agrees with G2 adjoint branching to a short-root sl2 subalgebra. All labels in the table are highest weights, rather than dimensions.

Write the simple roots of the G2 root system as $\alpha$ short and $\beta$ long. The Long-root A2 subsystem of G2 has simple rootsIf ${\omega }_1,{\omega }_2$ are its fundamental weights, then the $A_2$ root-weight relations giveFor the $A_2$-dominant weight $\lambda =a{\omega }_1+b{\omega }_2$, therefore,The dominant weight inequalities for $G_2$ are thus equivalent tobecause $A_2$-dominance already gives $a,b\ge 0$.