Root systems B_2 and C_2 from matrix Lie algebras

The equations defining both matrix spaces are closed under commutators, and [L2] makes the resulting Lie algebras semisimple. Their displayed diagonal subalgebras are abelian. Choose $H_B=\mathrm{diag}(1,2,0,-2,-1)$ and $H_C=\mathrm{diag}(1,2,-1,-2)$; each has pairwise distinct diagonal entries. If $X$ normalizes the corresponding diagonal subalgebra, then $[X,H_B]$ or $[X,H_C]$ is diagonal, but a commutator with a diagonal matrix has zero diagonal and therefore vanishes. The matrix-unit bracket formula then makes $X$ diagonal, and the defining form equation places it in ${\mathfrak{h}}_B$ or ${\mathfrak{h}}_C$. Thus each displayed subalgebra is nilpotent and self-normalizing, hence Cartan by [L1].

For ${\mathfrak{s}\mathfrak{o}}_5(\mathbb{C})$, put $a^{\mathrm{\prime }}=6-a$. Its zero-weight space is ${\mathfrak{h}}_B$, and its nonzero weight spaces are spanned by the nonzero vectors $E_{ab}-E_{b^{\mathrm{\prime }}{a}^{\mathrm{\prime }}}$ with $a\mathrm{\ne }b$ and $b\mathrm{\ne }{a}^{\mathrm{\prime }}$. Each is an ${\mathrm{ad}}_H$-eigenvector of weight $H_{aa}-H_{bb}$ because $H_{a^{\mathrm{\prime }}{a}^{\mathrm{\prime }}}=-H_{aa}$ and $H_{b^{\mathrm{\prime }}{b}^{\mathrm{\prime }}}=-H_{bb}$. Listing these weights gives $\pm {\epsilon }_1,\pm {\epsilon }_2,\pm ({\epsilon }_1+{\epsilon }_2),\pm ({\epsilon }_1-{\epsilon }_2)$; the would-be weights $\pm 2{\epsilon }_i$ correspond to $b=a^{\mathrm{\prime }}$, where the displayed vector is zero. Thus there are exactly eight roots, the set $B_2$.

For ${\mathfrak{s}\mathfrak{p}}_4(\mathbb{C})$, the form-compatible root vectors in the $a$-blocks are $E_{12}-E_{43}$ and $E_{21}-E_{34}$, of weights $\pm ({\eta }_1-{\eta }_2)$. The symmetric $b$-block gives $E_{13},E_{24},E_{14}+E_{23}$ of weights $2{\eta }_1,2{\eta }_2,{\eta }_1+{\eta }_2$, and the symmetric $c$-block gives their three negative weights. Hence the roots are $\pm 2{\eta }_1,\pm 2{\eta }_2,\pm ({\eta }_1+{\eta }_2),\pm ({\eta }_1-{\eta }_2)$ — exactly eight roots, the set $C_2$.

By [L3], the two root sets computed in steps 2.1 and 2.2 are reduced crystallographic root systems. The linear map $\phi ({\epsilon }_1)={\eta }_1+{\eta }_2$, $\phi ({\epsilon }_2)={\eta }_1-{\eta }_2$ carries the eight vectors of ${\mathrm{\Phi }}_B$ bijectively onto those of ${\mathrm{\Phi }}_C$. Its two image basis vectors are orthogonal of squared length $2$, so $(\phi u,\phi v)=2(u,v)$ for all $u,v$; the common factor cancels from every Cartan integer. Thus [L3] makes $\phi$ a root-system isomorphism $B_2\cong C_2$.