Split Cartan subalgebras of classical matrix Lie algebras

$\mathfrak{h}$ is abelian, hence nilpotent, and the linear map sending a diagonal matrix to its diagonal vector is an isomorphism of $\mathfrak{h}$ with the sum-zero hyperplane of ${\mathbb{C}}^n$ (respectively with ${\mathbb{C}}^n$ in the non-special-linear cases); the relevant dimensions are $n-1$ for ${\mathfrak{s}\mathfrak{l}}_n$ and $n$ otherwise.

Choose $H_0\in \mathfrak{h}$ whose full ambient diagonal entries are pairwise distinct: in ${\mathfrak{s}\mathfrak{l}}_n$ take $h_i=i-(n+1)\mathrm{/}2$; in the even symplectic and orthogonal cases take the diagonal entries $1,\dots ,n,-1,\dots ,-n$; and in the odd orthogonal case insert $0$ before those $2n$ entries. If $X\in N_{\mathfrak{g}}(\mathfrak{h})$, then $[X,H_0]\in \mathfrak{h}$ is diagonal. On the other hand every diagonal entry of a commutator with a diagonal matrix is zero, so $[X,H_0]=0$. Its $(a,b)$ entry is $(H_{0,bb}-H_{0,aa})X_{ab}$; distinctness therefore makes every off-diagonal entry of $X$ vanish. Intersecting the ambient diagonal matrices with the defining trace or form-preservation equations in [L1] gives exactly $\mathfrak{h}$. Thus $N_{\mathfrak{g}}(\mathfrak{h})=\mathfrak{h}$, and consequently $C_{\mathfrak{g}}(\mathfrak{h})=\mathfrak{h}$ as well.

The adjoint action of $\mathfrak{h}$ on the ambient matrix algebra is simultaneously diagonalizable: the matrix units $E_{ab}$ are common eigenvectors with eigenvalue $H_{aa}-H_{bb}$ by [L1]. Since $\mathfrak{g}$ is invariant under every ${\mathrm{ad}}_H$, their restrictions to $\mathfrak{g}$ are simultaneously diagonalizable, so $\mathfrak{h}$ is toral. Any toral subalgebra containing $\mathfrak{h}$ is abelian and hence lies in $C_{\mathfrak{g}}(\mathfrak{h})=\mathfrak{h}$ by step 1.2; therefore $\mathfrak{h}$ is maximal toral.

By steps 1.1 and 1.2 the subalgebra $\mathfrak{h}$ is nilpotent and equal to its normalizer, hence is a Cartan subalgebra; by step 2.1 it is maximal toral and equals its centralizer. This proves all the assertions.