Let A denote an R-algebra, so that A is a vector space over R and A×A->A (x, y)↦x·y. Then A is said to be alternative if, for all x, y element A, (x·y)·y = x·(y·y) (x·x)·y = x·(x·y). Here, vector multiplication x·y is assumed to be bilinear. The associator (x, y, z) is an alternating function, and the subalgebra generated by two elements is associative.
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