Proposition 2

Let g be a Lie algebra, a an ideal (resp. a characteristic ideal) of g and b a characteristic ideal of a. Then b is an ideal (resp. a characteristic ideal) of g.

Every inner derivation (resp. every derivation) of g leaves a stable and induces on a a derivation and hence leaves b stable.

Let g be a Lie algebra. If a and b are ideals of g, a + b and a ∩ b are ideals of g.

Let a and b be two submodules of g. By an abuse of notation, the submodule of g generated by the elements of the form [x, y] (x ∈ a, y ∈ b) is denoted by [a, b]. We have [a, b] = [b, a] by identity (3). If z ∈ g, [z, a], or [a, z], denotes the submodule [Kz, a] = (ad z)(a).