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The diagonal Cartan subalgebra consists ofLet ${\epsilon }_i\in {\mathfrak{t}}^{\ast }$ read off $t_i$. The roots arethe D3 root system.

A Cartan subalgebra of a complex semisimple Lie algebra is a maximal abelian subalgebra consisting of semisimple elements.

The weights of $M_{\lambda }$ lie below $\lambda$ in the positive-root order, and every weight space is finite-dimensional. A nonzero submodule $V$ is stable under the Cartan subalgebra, so it is a direct sum of its weight spaces. Choose a maximal weight $\mu$ occurring in $V$. Every positive-root operator would raise its weight; maximality therefore makes it kill any nonzero $v\in V_{\mu }$. Thus $v$ is a singular vector.

The Casimir element is central and acts throughout $M_{\lambda }$ byThe same element acts on the highest-weight vector $v$ of weight $\mu$ byBoth are the action of one operator on the same module, so the scalars agree and

Choose a short simple root ${\alpha }_1$ and a long simple root ${\alpha }_2$, with $|{\alpha }_2{|}^2=3|{\alpha }_1{|}^2$ and angle $150^{\circ }$. The six positive roots of the G2 root system areand their negatives complete the two concentric hexagons of short and long roots. The fundamental weights areso ${\omega }_1$ is itself a short root and ${\omega }_2$ is the highest root.

The seven-dimensional representation $L_{{\omega }_1}$ has weight seteach with weight multiplicity one. Their positive heights are $1,2,3$, so the q-character of a highest-weight representation isThis is one ${\mathfrak{sl}}_2$ weight string, hence

The representation $L_{{\omega }_2}$ is the fourteen-dimensional Adjoint representation. Its nonzero weights are the twelve roots, and its zero-weight space is the two-dimensional Cartan subalgebra. The positive root heights are $1,1,2,3,4,5$, soSplitting this into ordinary ${\mathfrak{sl}}_2$ strings gives

A real symmetric bilinear form is positive-definite when $B(v,v)>0$ for every nonzero vector. It defines an inner product and the associated norm $\sqrt{B(v,v)}$. Symmetry is part of this definition: positive diagonal values by themselves do not imply symmetry.^[ref]

The identity matrix $A=\mathrm{diag}(1,1)$ gives $B(v,w)=v_1{w}_1+v_2{w}_2$ and $B(v,v)=v_1^2+v_2^2$. For $v=(3,4)$ this is $25$, giving norm $5$. A weighted example, $A=\mathrm{diag}(4,1)$, gives norm $\sqrt{4x^2+y^2}$: the first coordinate costs twice as much in length. Such inner products let geometry assign different weights to different directions.

Every positive-definite bilinear form is nondegenerate: a vector in its radical of a bilinear form has $B(v,v)=0$, so positivity forces $v=0$. The converse fails, as $\mathrm{diag}(1,-1)$ shows. Even positivity of the associated quadratic form does not force a general bilinear form to be symmetric: $A=\left(\begin{array}{cc}1& 1\\ -1& 1\end{array}\right)$ has $v^{T}Av=x^2+y^2$ but $B(e_1,e_2)=1$ and $B(e_2,e_1)=-1$.

The complex Killing form is not positive-definite on the entire complex Cartan subalgebra; its restriction to the Euclidean subspace of a Cartan subalgebra is a real positive-definite form.

The rank is the dimension of a Cartan subalgebra, equivalently the dimension of the real span of its root system.

For $t$ in a Cartan subalgebra $\mathfrak{t}$, the root-space decomposition givesThus $t$ is regular exactly when $\alpha (t)\ne 0$ for every root $\alpha$.

For root spaces ${\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{\beta }$, the Jacobi identity gives $[{\mathfrak{g}}_{\alpha },{\mathfrak{g}}_{\beta }]\subset {\mathfrak{g}}_{\alpha +\beta }$ when $\alpha +\beta \ne 0$. If the bracket is nonzero, that nonzero weight must be a root. When $\beta =-\alpha$, the bracket instead lies in the Cartan subalgebra and can be nonzero; zero is not a root. Thus the common nonzero-bracket implication requires the opposite-root exception.

A root is an individual nonzero vector in a root system. For a complex semisimple Lie algebra, it is a nonzero weight of the Adjoint representation restricted to a Cartan subalgebra; its root space consists of vectors $x$ satisfying $[h,x]=\alpha (h)x$ for all $h$ in that subalgebra.

For a root $\alpha$, its root space is the simultaneous eigenspaceof the Adjoint representation of the Cartan subalgebra.

For a Cartan subalgebra $H$ of a finite-dimensional complex semisimple Lie algebra $L$, $H$ is abelian and $C_L(H)=H$, so it is maximal among abelian subalgebras. Starting from the nilpotent self-normalizing definition, the generalized-weight decomposition for a nilpotent Lie algebra and the Engel theorem show that the zero generalized weight space of its adjoint action is exactly $H$. Invariance of the Killing form makes $H$ orthogonal to every nonzero generalized weight space. The Lie theorem gives $B([H,H],H)=0$, and nondegeneracy then forces $[H,H]=0$. An element commuting with $H$ normalizes it, hence belongs to $H$.

A semisimple Lie algebra over a field of characteristic zero is split if it has a Cartan subalgebra whose adjoint action is simultaneously diagonalizable over that field. Its root-space decomposition then needs no extension of scalars. The sl2R Lie algebra is split, with the real diagonal Cartan subalgebra. The real special unitary Lie algebra $\mathfrak{su}(2)$ is not: its adjoint operators are skew-adjoint for the positive definite inner product obtained by negating its Killing form. Thus a split adjoint operator would have to be zero.

For $V={\mathbb{C}}^n$, $n\ge 3$, the tensor product of Lie algebra representations for the special linear Lie algebra splits as displayed. The exterior product map $v\otimes (u\wedge w)\mapsto v\wedge u\wedge w$ has an equivariant right inverse obtained by dividing its three-term alternating insertion by three. Its kernel is the Schur functor of shape $(2,1)$, with highest weight ${\omega }_1+{\omega }_2$ and dimension $n(n^2-1)/3$. The other summand has highest weight ${\omega }_3$ for $n\ge 4$ and is trivial for $n=3$.

The weight space of $2{\epsilon }_i+{\epsilon }_j$ has dimension one when $i\ne j$. The weight space of ${\epsilon }_i+{\epsilon }_j+{\epsilon }_k$ has dimension three for distinct indices. Restricting these functionals to the traceless diagonal Cartan subalgebra creates no additional coincidences; for $n=3$ the latter weight is zero.

A toral subalgebra of a Lie algebra $\mathfrak{g}$ is a Lie subalgebra $\mathfrak{a}$ such that $\mathrm{ad}a$ is diagonalizable for every $a\in \mathfrak{a}$. Here diagonalizability is over the stated ground field, and refers to the Adjoint representation. In arguments that use a simultaneous weight decomposition, commutativity of $\mathfrak{a}$ is stated as well. For a semisimple Lie algebra over an algebraically closed field of characteristic zero, Cartan subalgebras are examples. Semisimplicity on the defining representation of a symplectic Lie algebra follows from adjoint semisimplicity in the symplectic Lie algebra.

Choosing a positive system of a root system for a complex semisimple Lie algebra divides its root spaces into negative and positive ones. Their respective sums ${\mathfrak{n}}^{-}$ and ${\mathfrak{n}}^{+}$, together with the Cartan subalgebra $\mathfrak{h}$, give this direct sum of vector spaces. Both ${\mathfrak{n}}^{\pm }$ are nilpotent Lie algebras, and $\mathfrak{h}\oplus {\mathfrak{n}}^{+}$ is a Borel subalgebra. The Poincaré-Birkhoff-Witt theorem turns this into an ordered vector-space decomposition of the universal enveloping algebra, useful for building Verma modules.

A weight basis of a Lie algebra representation is a basis consisting of weight vectors for a chosen Cartan subalgebra. When the Cartan action is diagonalizable, choose a basis separately in every weight space and combine them. This makes weight multiplicities directly visible and gives diagonal matrices for Cartan elements.

A diagram placing the weights of a representation in the real span of its weight lattice, recording each weight multiplicity. Coincident weights must retain their multiplicity: for an Adjoint representation the zero-weight multiplicity is the dimension of its Cartan subalgebra.

A coordinate matrix expressing restriction of weights to the Cartan subalgebra of a Lie subalgebra. With Dynkin labels as column vectors, its rows encode the subalgebra's coroots in terms of the original coroots. It need not be an idempotent projection operator: an equal-rank embedding can give an invertible matrix. Applying it to the full weight system, including weight multiplicities, permits reconstruction of the restricted irreducible representations.

A weight-space decomposition expresses a Lie algebra representation as a direct sum of simultaneous eigenspaces for a Cartan subalgebra. Each summand is the weight space for its weight. In complex finite-dimensional representations of a semisimple Lie algebra, the simple coroots act as commuting diagonalizable operators, giving this decomposition.

The weight system of a finite-dimensional Lie algebra representation is its collection of weights with their weight multiplicities. Restriction to a Lie subalgebra projects each weight to the smaller Cartan subalgebra, retaining and combining weight multiplicities. Grouping the resulting weights into irreducible weight systems computes the branching rule.