Alternate Algebra

Let $A$ denote an $\Bbb{R}$-Algebra, so that $A$ is a Vector Space over $R$ and

$$A\times A\to A$$ (1)

$$(x,y)\mapsto x\cdot y.$$ (2)

Then $A$ is said to be alternate if, for all $x,y\in A,$

$$(x\cdot y)\cdot y=x\cdot(y\cdot y)$$ (3)

$$(x\cdot x)\cdot y=x\cdot(x\cdot y).$$ (4)

Here, Vector Multiplication $x\cdot y$ is assumed to be Bilinear.

References

Finch, S. ``Zero Structures in Real Algebras.'' http://www.mathsoft.com/asolve/zerodiv/zerodiv.html.

Schafer, R. D. An Introduction to Non-Associative Algebras. New York: Dover, 1995.