Cartan subalgebra

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over an algebraically closed field of characteristic $0$ (typically $\mathbb{C}$).

Definition. A subalgebra $\mathfrak{h}\subset \mathfrak{g}$ is a Cartan subalgebra if:

1. $\mathfrak{h}$ is nilpotent as a Lie algebra, and
2. $\mathfrak{h}$ is self-normalizing in $\mathfrak{g}$, i.e. $N_{\mathfrak{g}}(\mathfrak{h})=\mathfrak{h}$ (compare the self-normalizing lemma ).

Here $N_{\mathfrak{g}}(\mathfrak{h})=\{X\in\mathfrak{g}:[X,\mathfrak{h}]\subset \mathfrak{h}\}$ is the normalizer Lie subalgebra.

Semisimple refinement. If $\mathfrak{g}$ is semisimple , Cartan subalgebras can be characterized as maximal abelian subalgebras consisting of semisimple endomorphisms in the adjoint representation. With such an $\mathfrak{h}$, $\mathfrak{g}$ admits the root space decomposition

where $\Phi\subset \mathfrak{h}^*$ is the root set and $\mathfrak{g}_\alpha$ are the root spaces .

Motivation. Cartan subalgebras are the “coordinate axes” for semisimple structure: weights of representations live in $\mathfrak{h}^*$ (see weights in the dual Cartan ), and the choice of $\mathfrak{h}$ underlies Dynkin-diagram data such as the Cartan matrix .