a finite-dimensional vector space V) over an algebraically closed field, then any Cartan subalgebra of is the centralizer of a maximal toral subalgebra of . If is semisimple and the field has characteristic zero, then a maximal toral subalgebra is self-normalizing, and so is equal to the associated Cartan subalgebra. If in addition is semisimple, then the adjoint representation presents as a linear Lie algebra, so that a subalgebra of is Cartan if and only if it is a maximal toral subalgebra. Examples Any nilpotent Lie algebra is its own Cartan subalgebra. A Cartan subalgebra of gln, the Lie algebra of n×n matrices over a field, is the algebra of all diagonal matrices. For the special Lie algebra of traceless matrices , it has the Cartan subalgebra where For example, in the Cartan subalgebra is the subalgebra of matrices with Lie bracket given by the matrix commutator. 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 (or, since it is normalized by diagonal matrices). Cartan subalgebras of semisimple Lie algebras For finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic 0, a Cartan subalgebra has the following properties: is abelian, For the adjoint representation , the image consists of semisimple operators (i.e., diagonalizable matrices). (As noted earlier, a Cartan subalgebra can in fact be characterized as a subalgebra that is maximal among those having the above two properties.) These two properties say that the operators in are simultaneously diagonalizable and that there is a direct sum decomposition of as where . Let . Then is a root system and, moreover, ; i.e., the centralizer of coincides with . The above decomposition can then be written as: As it turns out, for each , has dimension one and so: . See also Semisimple_Lie algebra#Structure for further information. Decomposing representations