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For B3 root system simple roots $e_1-e_2,e_2-e_3,e_3$, choose the D3 root system simple roots in chain order $e_2-e_3,e_1-e_2,e_2+e_3$. The Dynkin labels project by $(a,b,c)\mapsto (b,a,b+c)$. This separates the vector weights $\{0,\pm e_i\}$ into the six-dimensional exterior square of the defining SU(4) representation and a singlet representation. The spinor cube vertices $\frac{1}{2}(\pm e_1\pm e_2\pm e_3)$ split into its defining representation and its dual representation.

Choose an orthonormal basis $e_1,e_2,e_3$ in the real Euclidean space containing the root system, and take simple rootsTheir squared lengths are $2,2,1$. Use the Cartan matrix convention $A_{ij}=2({\alpha }_i,{\alpha }_j)/({\alpha }_j,{\alpha }_j)$, with the denominator indexed by the column. Direct inner products giveThe Dynkin diagram of the B3 root system is a three-node chain with a single bond from node 1 to node 2 and a double bond from node 2 to the short node 3; the arrow points toward node 3. The A3 root system has a chain of three nodes joined by single bonds. A convention putting the row length in the denominator transposes the first displayed Cartan matrix.

For the Extended Dynkin diagram, the highest root is $\theta =e_1+e_2={\alpha }_1+2{\alpha }_2+2{\alpha }_3$. Add ${\alpha }_0=-\theta$. Its only nonzero off-diagonal inner product among these simple roots is $({\alpha }_0,{\alpha }_2)=-1$. Deleting the short node 3 leaves the chain $0-2-1$, with three equal-length roots, which is $A_3$.

More concretely, the long roots $\pm e_i\pm e_j$ form the D3 root system, which is isomorphic to the A3 root system. The corresponding root spaces and Cartan subalgebra generatefor the compact real forms. At the matrix level the inclusion fixes one coordinate in seven-dimensional space. This establishes the required Lie subalgebra inclusion; the associated global groups require the distinction explained in d.

In the convention of a, the fundamental weights areFor a weight $w=(w_1,w_2,w_3)$ in this orthonormal coordinate system, its Dynkin labels areThese labels, rather than the orthonormal coordinates, are used below and in the projection in c.

The highest weight $(1,0,0)$ is $e_1$. Successive subtraction of the simple roots gives the complete stringThe two consecutive short-root steps are allowed because $e_3$ has coroot pairing two with ${\alpha }_3$. The weight system of this seven-dimensional vector representation is thereforeAll seven weight multiplicities are one, including the zero weight.

The highest weight $(0,0,1)$ is ${\omega }_3$. The Weyl group of the B3 root system acts by signed coordinate permutations. Its orbit is the eight cube vertices $w=(s_1{e}_1+s_2{e}_2+s_3{e}_3)/2$, with independent $s_i=\pm 1$. Starting from $(+,+,+)$, lowering by ${\alpha }_3$, then ${\alpha }_2$, and then either ${\alpha }_1$ or ${\alpha }_3$ generates the middle branches; further lowering reaches all remaining sign choices. Thus the eight-dimensional spin representation hasThe stipulated unit weight multiplicities give dimensions seven and eight by counting these lists.

With the Euclidean inner product used above, the simple coroots areIndeed, if a positive root is $\beta ={\sum }_i{n}_i{\alpha }_i$, thenwhose coefficients are nonnegative. The negative roots give the negatives of these combinations, so the displayed coroots form a fundamental system of a root system for the dual root system.

Duality reverses root lengths. The dual of $C_3$ is therefore the B3 root system: its Dynkin diagram is again a three-node chain with a double final edge, but its arrow points toward the now-short root ${\alpha }_3^{\vee }$.

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 compact group of real seven-by-seven orthogonal matrices with determinant one. Its Lie algebra has B3 root system. Its simply connected double cover is the Spin group $\mathrm{Spin}(7)$; the eight-dimensional spin representation belongs to that cover and does not descend to SO(7).