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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?

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  • $\begingroup$ What do you mean it contains "$n-1$" elements? You mean that is the dimension? $\endgroup$ Commented Aug 6, 2015 at 8:12
  • $\begingroup$ "N-1" is the number of elements in Cartan subalgebra. Number of commuting elements of a given Lie algebra, if you will. $\endgroup$
    – Kosm
    Commented Aug 7, 2015 at 3:27
  • $\begingroup$ No, neither of those statements are correct. Any non-zero subalgebra has an infinite number of elements. As explained by Dietrich Burde in the answer, the $n-1$ is the rank of the root system and the dimension of the Cartan subalgebra. $\endgroup$ Commented Aug 7, 2015 at 5:03
  • $\begingroup$ Let me explain what I mean in an example. Cartan subalgebra of SU(3) has two commuting generators: $\lambda_3$ and $\lambda_8$, in terms of Gell-Mann matrices. $\endgroup$
    – Kosm
    Commented Aug 7, 2015 at 5:28
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    $\begingroup$ I am fairly certain you meant dimension (i.e. minimal number of generators), as already pointed out. $\endgroup$ Commented Aug 7, 2015 at 5:30

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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 $SO(2n)$.

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