A possibly non-associative algebra is *flexible* if it satisfies the identity $$(xy)x=x(yx).$$ This is clearly a very weak form of associativity —and obviously an associative algebra is flexible— but it is also a weak form of commutativity —a commutative algebra is clearly flexible— and in practice it plays both of these roles.

Flexibility is such a weak condition that it is difficult to imagine getting much out of it, but then one reads the works of Albert, Schafer, Jacobson and several others, and sees it put to use with such extraordinary artistry and skill that one ends up reflecting on the excessiveness of our standard hypotheses of commutativity and associativity :-)

But I have not found so far an explanation for the choice of the term:

Do you know why flexible algebras are called flexible?