Remark

(1) The algebra {0} is semi-simple. An algebra of dimension 1 or 2 is not semi-simple (cf. § 5, no. 1, Example 1). There exist semi-simple algebras of dimension 3 (cf. no. 7).

(2) A semi-simple algebra has zero centre and hence its adjoint representation is faithful.

(3) If $g_1, \ldots, g_n$ are semi-simple, $g = g_1 \times \cdots \times g_n$ is semi-simple: for if a is a commutative ideal of g, the projections of a onto $g_1, \ldots, g_n$ reduce to {0}.