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$\DeclareMathOperator\rad{rad}$1) is a special case of the following statement for finite dimensional algebras: Let $A$ and $B$ be a finite dimensional algebras over a field $K$ such that $A/\rad(A)$ and $B/\rad(B)$ are isomorphic to a direct product of matrix algebras over $K$. When $e_i$ and $e_i'$ are pariwise orthogonal primitive idempotents which sum to 1 for $A$ and $B$ respectively, then $e_i \otimes_K e_i'$ are pairwise orthogonal primitive idempotents which sum to 1 for $A \otimes_K B$.