Remark

If a Lie algebra g can be decomposed as a product a × b of a commutative Lie algebra a and a semi-simple Lie algebra b, this decomposition is unique. More precisely, the centre of g is equal to the product of the centres of a and b and is hence equal to a. And $\mathscr{D}g = \mathscr{D}a \times \mathscr{D}b = b$.