Proposition 6

Let L be the diagonalizable subalgebra of End$_D(V)$ associated with a direct sum decomposition $V =\oplus_{i\in I}V_i$.

a) The algebra L is maximal among the diagonalizable subalgebras of the K-algebra End$_D(V)$ if and only if every $V_i$ has dimension 1 over D.

b) The algebra L is a maximal commutative subalgebra of End$_D(V)$ if and only if we have D = K and every $V_i$ has dimension 1 over K.

If every vector space $V_i$ has dimension 1 over D, then we have

[L : K] = Card(I) = dim$_D(V)$,

so L is maximal among the diagonalizable subalgebras of End$_D(V)$. In the opposite case, there exists an index $j\in I$ such that dim$_D(V_j)\geqslant 2$. Choose two nonzero linear subspaces $V'_j$ and $V''_j$ of $V_j$ with direct sum $V_j$. The diagonalizable subalgebra of End$_D(V)$ associated with the direct sum decomposition $V = (\oplus_{i\in I-\{j\}}V_i)\oplus V'_j\oplus V''_j$ contains L and is not equal to L; assertion a) follows.

The commutant $L'$ of L in End$_D(V)$ consists of the endomorphisms of the form $(x_i)\mapsto (u_i(x_i))$, with $(u_i)\in \prod_{i\in I}$ End$_D(V_i)$. The algebra L is a maximal commutative subalgebra of End$_D(V)$ if and only if we have $L = L'$ (VIII, p. 261, Lemma 3, a)). This relation is therefore equivalent to “End$_D(V_i) = K$ for every $i\in$ I”; assertion b) follows.