Definition 6

Let

be an extension of b by a and n its kernel. This extension is called inessential (resp. trivial) if there exists a subalgebra (resp. an ideal) of g supplementary to n in g. This extension is called central if n is contained in the centre of g.

If the extension is trivial, let m be an ideal of g supplementary to n in g. Then (cf. no. 1) g is canonically identified with the Lie algebra m × n and hence with the Lie algebra a × b. Conversely, let a and b be two Lie algebras; then a × b is a trivial extension of a by b.

An inessential central extension is trivial. For let g be a Lie algebra, n an ideal of g contained in the centre of g and m a subalgebra of g supplementary to n in g. Then [m, g] = [m, m] + [m, n] = [m, m] ⊂ m and hence m is an ideal of g.