Let g be a finite-dimensional Lie algebra over some field k. A subalgebra h of g is called a Cartan subalgebra if it is nilpotent and equal to its normalizer, which is the set of those elements x element g such that [x, h] subset h. It follows from the definition that if g is nilpotent, then g itself is a Cartan subalgebra of g. On the other hand, let g be the Lie algebra of all endomorphisms of k^n (for some natural number n), with [f, g] = f°g - g°f. Then the set of all endomorphisms f of k^n of the form f(x_1, ..., x_n) = (λ_1 x_1, ..., λ_n x_n) is a Cartan subalgebra of g.

Cartan algebra | subalgebra