Proposition 6

Let E, F be two non-zero algebras over a commutative field, K, C (resp. D) a subalgebra of E (resp. F) and C' (resp. D') the centralizer of C in E (resp. D in F). Then the centralizer of C \otimesK D in E \otimes_K F is C' \otimes_K D'.*

It all reduces to verifying that an element $z = \sum_i x_i \otimes y_i$ of the centralizer of $C \otimes_K D$ ($x_i \in F, y_i \in F$) belongs to $C' \otimes_K D'$; we know that

(II, § 7, no. 7, Corollary to Proposition 14). The $y_i$ may be assumed to be linearly independent over K; for all $x \in C$, of necessity $(x \otimes e')z = z(x \otimes e')$, that is $\sum_i (xx_i - x_ix) \otimes y_i = 0$, whence $xx_i = x_ix$ for all $i$ (II, § 3, no. 7, Corollary 1 to Proposition 7); hence of necessity $x_i \in C'$ for all $i$ and therefore $z \in C' \otimes_K F$; it can similarly be shown that $z \in E \otimes_K D'$, whence the proposition.