Corollary

(a) Every finite product of reductive algebras is a reductive algebra. (b) If g is a reductive Lie algebra of centre c, every ideal of g is a direct factor, the product of its intersections with c and $\mathscr{D}g$, and is a reductive Lie algebra. (c) Every quotient of a reductive Lie algebra is a reductive Lie algebra.

Assertion (a) follows for example from condition (c) of Proposition 5.

Suppose that g is reductive. Let a be an ideal of g. Since the adjoint representation of g is semi-simple, a has a supplementary ideal b and g is identified with a × b. For all $x \in g$, let $\rho(x)$ be the restriction of $\mathrm{ad}_g x$ to a. Then $\rho$ is a semi-simple representation of g which is zero on b and defines on passing to the quotient the adjoint representation on a. Hence a is reductive. Similarly, g/a and b, which are isomorphic, are reductive. Finally, let d, d' be the centres of a and b; then $a = d \times \mathscr{D}a, b = d' \times \mathscr{D}b, d \times d' = c, \mathscr{D}a \times \mathscr{D}b = \mathscr{D}g$; hence $a = (a \cap c) + (a \cap \mathscr{D}g)$.