Corollary 2

The assumptions and notations being those of cor. 1, suppose in addition that the algebra B is commutative. Then all its elements are semisimple (normal) endomorphisms of E. When A = K (resp. A = K(i)), E is the direct sum of simple B-submodules pairwise orthogonal, which are vector spaces of dimension 1 or 2 (resp. 1) over A.

The first assertion follows from chap. VIII, § 9, No. 1, prop. 2, since B is semisimple. On the other hand every simple B-module is isomorphic to a B-module of the form B/m, where m is a maximal ideal of B, hence to a commutative field L of finite degree over A; when A = K (resp. A = K(i)), the field L is necessarily isomorphic to K or K(i) (resp. K(i)), since K(i) is algebraically closed (chap. VI, § 2, No. 6, th. 3).