Examples
- Any nilpotent Lie algebra is its own Cartan subalgebra.
- A Cartan subalgebra of the Lie algebra of n×n matrices over a field is the algebra of all diagonal matrices.
- The Lie algebra sl2(R) of 2 by 2 matrices of trace 0 has two non-conjugate Cartan subalgebras.
- The dimension of a Cartan subalgebra is not in general the maximal dimension of an abelian subalgebra, even for complex simple Lie algebras. For example, the Lie algebra sl2n(C) of 2n by 2n matrices of trace 0 has a Cartan subalgebra of rank 2n−1 but has a maximal abelian subalgebra of dimension n2 consisting of all matrices of the form with A any n by n matrix. One can directly see this abelian subalgebra is not a Cartan subalgebra, since it is contained in the nilpotent algebra of strictly upper triangular matrices (which is also not a Cartan subalgebra since it is normalized by diagonal matrices).
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