Definition 1

Let $\mathfrak{g}$ be a Lie algebra. A Cartan subalgebra of $\mathfrak{g}$ is a nilpotent subalgebra of $\mathfrak{g}$ equal to its own normalizer.

Later we shall obtain the following results:

1) if $k$ is infinite, $\mathfrak{g}$ has Cartan subalgebras (no. 3, Cor. 1 of Th. 1);

2) if $k$ is of characteristic zero, all Cartan subalgebras of $\mathfrak{g}$ have the same dimension (§3, no. 3, Th. 2);

3) if $k$ is algebraically closed and of characteristic 0, all Cartan subalgebras of $\mathfrak{g}$ are conjugate under the group of elementary automorphisms of $\mathfrak{g}($§3, no. 2, Th. 1).