Number of Cartan subalgebra elements

Question

For SU(n) Lie algebra, the Cartan subalgebra contains "n - 1" elements. What are numbers of elements in SO(n) [maybe separately for SO(2n) and SO(2n+1)?] and Sp(n) Cartan subalgebras?

Answer 1

I suppose you are interested in the rank of these simple Lie algebras. The rank of a Lie algebra of characteristic zero is given by the dimension of a Cartan subalgebra (all Cartan subalgebras have the same dimension in this case). The classification of complex simple Lie algebras takes the rank as an index, i.e., we have rank $n$ for the simple Lie algebras of type $A_n$, $B_n$, $C_n$ and $D_n$. Here type $A_n$ corresponds to $SL(n+1)$ (over the real numbers $SU(n+1)$), and $B_n$ to $SO(2n+1)$, $C_n$ to $Sp(2n)$ and $D_n$ to $SO(2n)$.