I have $A$ an associative algebra and $B$ at least an alternative algebra. Is there a sufficient condition on $A$ or $B$ to have $A \otimes B$ an alternative algebra?



Alternativity on $A \otimes B$
The monograph “Alternative Loop Rings” by Goodaire, Jespers and Polcino Milies (North Holland Mathematics Studies **184**, 1996) contains, in chapter I (“Alternative Rings”), §5 (“Tensor Products”), the following proposition (5.13; I'm changing the notation to match yours):



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> Let $B$ be an alternative algebra over a field $F$ and suppose $A$ is a commutative associative algebra over $F$. Then the tensor product $A\otimes\_F B$ is alternative.
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So $A$ being commutative is a sufficient condition.


