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The $B_3$ roots are $\pm {\epsilon }_i\pm {\epsilon }_j$ for $1\le i<j\le 3$ and $\pm {\epsilon }_i$ for $1\le i\le 3$. Its root basis ${\alpha }_1={\epsilon }_1-{\epsilon }_2$, ${\alpha }_2={\epsilon }_2-{\epsilon }_3$, ${\alpha }_3={\epsilon }_3$ exhibits it as the dual root system of the C3 root system. With $A_{ij}=⟨{\alpha }_i,{\alpha }_j^{\vee }⟩$, its Cartan matrix isThe fundamental representation of highest weight $[1,0,0]$ is the seven-dimensional vector representation, with weights $\pm {\epsilon }_1,\pm {\epsilon }_2,\pm {\epsilon }_3,0$.

For the C3 root system, take ${\alpha }_1=e_1-e_2$, ${\alpha }_2=e_2-e_3$, ${\alpha }_3=2e_3$. With columns indexing coroots, the Cartan matrix isThe convention with rows indexing coroots gives $A^T$. Both describe a double bond between the short root ${\alpha }_2$ and long root ${\alpha }_3$, with arrow toward ${\alpha }_2$.

The double-bond arrow points to the middle node in the PDF. Since $\Vert \alpha \Vert =1$ and $\Vert \beta \Vert =\sqrt{2}$, the short root $\alpha$ must occupy that middle node and the long root $\beta$ the left node. The right node $\gamma$, joined to $\alpha$ by a single bond, is another short root. Thus the diagram is C3 root system type, labelled as follows.

The bonds require $\gamma \cdot \beta =0$, $\gamma \cdot \alpha =-1/2$ and $\Vert \gamma {\Vert }^2=1$. Writing $\gamma =(a,b,c)$ gives $a=-1/2$, $b=a$, and $c^2=1/2$. One choice is thereforeThe opposite choice of the last coordinate reflects the entire realization across the first two coordinate axes' plane and is equally valid.

To enumerate exhaustively, set $f_1=(0,0,1)$, $f_2=(1,1,0)/\sqrt{2}$ and $f_3=(-1,1,0)/\sqrt{2}$. These are orthonormal, andThe complete C3 root system is $\{\pm \sqrt{2}{f}_i,(\pm f_i\pm f_j)/\sqrt{2}:i<j\}$, with eighteen roots. Its nine roots with nonnegative coefficients in this simple basis are exactlyThere are six short and three long positive roots, and the other nine roots are their negatives.

The single-bond rank-two diagram is A2 root system type. Three suitable positive simple pairs for its root subsystems areEach pair $p,q$ has squared lengths one and inner product $-1/2$, so its angle is $120$ degrees. Its plane meets the full root set in $\pm p,\pm q,\pm (p+q)$, with $p+q$ respectively $\alpha +\gamma$, $2\alpha +\beta +\gamma$ and $\alpha +\beta +\gamma$. These exhaust the roots of the plane, and their coordinates in the pair have one sign. Thus each pair is genuinely a simple basis of an $A_2$ subsystem.

The double-bond diagram is rank-two B2 root system type, equivalently $C_2$. Three suitable pairs, written short root first, areFor each pair $s,l$, $\Vert s{\Vert }^2=1$, $\Vert l{\Vert }^2=2$ and $s\cdot l=-1$. The remaining positive roots in its plane areTogether with their negatives these are precisely the eight plane roots, and all four positive roots have nonnegative integral pair-coordinates. Hence the displayed pairs are simple bases of the required subsystems. The rank-two subsystems of a C3 root system description explains these as the three coordinate-plane subsystems in the orthonormal $f_i$ realization; the chosen $A_2$ pairs lie in three different diagonal planes.

Here root systems are finite, reduced and crystallographic, as for complex semisimple Lie algebras. For a base of a root system $\Delta =\{{\alpha }_1,\dots ,{\alpha }_r\}$, the Dynkin diagram has one vertex for each simple root. Two distinct vertices have $a_{ij}{a}_{ji}$ bonds, with the arrow on a multiple bond pointing toward the shorter root. The classification of finite crystallographic Dynkin diagrams consists of the following connected diagrams; the integers describing arms count vertices away from the unique branching vertex.

This is classification up to diagram isomorphism and overall rescaling of the root system. It includes the conventional identifications $D_3\cong A_3$ and $B_1\cong A_1$ when those low-rank notations are used. The weighting here encodes root lengths and bonds; it is not the separate theory of vertex labels attached to nilpotent orbits. Dropping the crystallographic or reduced hypothesis changes the classification.

Precisely, a base of a root system is a vector-space basis $\Delta \subset \Phi$ such that each root has a unique expansion $\beta ={\sum }_i{n}_i{\alpha }_i$, with integer coefficients all nonnegative or all nonpositive. This defines the positive and negative roots. We use the column-coroot convention for the Cartan matrix:Some conventions transpose this matrix; specifying which index labels the coroot resolves that difference. The Weyl group is the subgroup of orthogonal transformations generated by the Weyl reflections

For the C3 root system, take an orthonormal basis $e_1,e_2,e_3$ of ${\mathbb{R}}^3$ andThere are six long roots of squared length $4$ and twelve short roots of squared length $2$. A base is ${\alpha }_1=e_1-e_2$, ${\alpha }_2=e_2-e_3$, ${\alpha }_3=2e_3$. The Cartan matrix convention for C3 above givesThus the double bond connects ${\alpha }_2$ to ${\alpha }_3$, with its arrow directed toward ${\alpha }_2$.

An explicit complex Lie algebra realizing this root system is the symplectic Lie algebraIt is closed under commutators, since the two identities $X^{T}J=-JX$, $Y^{T}J=-JY$ imply $[X,Y{]}^{T}J=-J[X,Y]$. Equivalently,so its dimension is $9+6+6=21$. A Cartan subalgebra isWith $E_{ab}$ denoting a matrix unit, the root spaces have generators $E_{ij}-E_{3+j,3+i}$ for $e_i-e_j$ ($i\ne j$), $E_{i,3+j}+E_{j,3+i}$ for $e_i+e_j$ ($i<j$), and $E_{i,3+i}$ for $2e_i$; the analogous lower-left generators give the negative roots. These are all eighteen root spaces, each one-dimensional, and together with $H$ they exhaust the algebra.

For an explicit semisimplicity check, let $I\ne 0$ be an ideal. Simultaneous diagonalization of $\mathrm{ad}H$ makes $I$ a sum of its intersections with $H$ and these root spaces. If it contains a nonzero $h\in H$, some root has $\alpha (h)\ne 0$, and $[h,L_{\alpha }]$ supplies a root vector in $I$. Thus $I$ always contains a root vector. Bracket it with its opposite root vector to obtain a nonzero coroot $h_{\alpha }\in I$. Whenever $(\alpha ,\beta )\ne 0$, bracketing $h_{\alpha }$ with either $L_{\beta }$ or $L_{-\beta }$ puts both root spaces into $I$. The graph of the roots with edges for nonzero inner products is connected: $2e_i$ connects to $e_i+e_j$, which connects to $2e_j$, and every short root connects to an axis root. Repeating the argument yields all root spaces and then all of $H$ from their brackets. Hence $I=L$: the constructed algebra is simple, and therefore semisimple.

The reflections in $e_1-e_2$ and $e_2-e_3$ exchange adjacent coordinates, while reflection in $2e_3$ changes the sign of the third coordinate. They generate every signed permutation. Conversely reflection in any displayed root is a signed permutation. ThusThe reflecting hyperplanes are $x_i=0$ and $x_i=\pm x_j$. A fundamental open Weyl chamber is $x_1>x_2>x_3>0$. Every regular point has three nonzero coordinates of distinct absolute values; its signs and the ordering of those absolute values specify exactly one of the $2^3\cdot 3!=48$ chambers. The closure of this chamber is generated by the rays through $(1,0,0)$, $(1,1,0)$ and $(1,1,1)$. This is the Weyl chamber geometry of C3.

The left panel retains the actual long and short root lengths. The right panel radially projects a hemisphere of the unit sphere, draws every visible reflecting great-circle arc, and highlights the chamber $x_1>x_2>x_3>0$. Each spherical triangle is the section of a three-dimensional chamber by the sphere; opposite triangles on the rear hemisphere supply the remaining chambers. Roots of a root system are normal to the chamber walls, rather than generally lying along their edges.

Write an element of the diagonal Cartan subalgebra asand define the linear functionals ${\epsilon }_i(t)=t_i$. The root-space decomposition of the symplectic Lie algebra ${\mathfrak{sp}}_6(\mathbb{C})$ then has the C3 root system

The root lattice is $Q={\sum }_i\mathbb{Z}{\alpha }^{(i)}$. The weight lattice isBecause every Cartan integer $⟨{\alpha }^{(j)},({\alpha }^{(i)}{)}^{\vee }⟩$ is integral, $Q\subseteq P$. The Dynkin labels of $\lambda$ are

There are three isomorphism classes of complex simple rank-three Lie algebras: types A3 root system$A_3$, B3 root system$B_3$, and C3 root system$C_3$. Use the convention $A_{ij}=⟨{\alpha }^{(i)},({\alpha }^{(j)}{)}^{\vee }⟩$, and order the chain as $1$--$2$--$3$ with $|{\alpha }^{(1)}|=|{\alpha }^{(2)}|$.

The Dynkin labels of a finite-dimensional irreducible representation are those of its highest weight. Thus $[1,0,0]$ means the fundamental representation $V({\omega }_1)$. The weights, written in Dynkin labels and in a lowering order, areFor $A_3$ these are the weights of the defining representation of ${\mathfrak{sl}}_4$; for $B_3$ they are $\{\pm e_1,\pm e_2,\pm e_3,0\}$ in the vector representation of ${\mathfrak{so}}_7$; for $C_3$ they are $\{\pm e_1,\pm e_2,\pm e_3\}$ in the defining representation of ${\mathfrak{sp}}_6$. Hence the requested dimensions are $4$, $7$, and $6$, respectively.

The C3 root system consists of $\pm 2e_i$ and $\pm e_i\pm e_j$ in ${\mathbb{R}}^3$. Its Weyl group is the signed symmetric group on three coordinates, of order $48$. The reflecting planes are $x_i=0$ and $x_i=\pm x_j$. The open Weyl chamber $x_1>x_2>x_3>0$ has closure generated by $(1,0,0)$, $(1,1,0)$ and $(1,1,1)$. Sign choices and absolute-coordinate orderings give its $48$ translates. On the unit sphere the chamber is a spherical triangle with angles $\pi /2$, $\pi /3$, $\pi /4$, and hence area $\pi /12$; the areas of all $48$ triangles sum to $4\pi$.