They are however alternative algebras (a weaker form of associativity).

In fact, this example shows that the hyperbolic quaternions are not even an alternative algebra.

The converse, however, is not true, in contrast to the situation in alternative algebras.

The associator of an alternative algebra is therefore alternating.

The split-octonions satisfy the Moufang identities and so form an alternative algebra.

Examples include all finite-dimensional alternative algebras, and the algebra of real 2-by-2 matrices.

In a unital alternative algebra, multiplicative inverses are unique whenever they exist.

It follows that expressions involving only two variables can be written without parenthesis unambiguously in an alternative algebra.

This alternative algebra is called algebra of physical space (APS).

Artin's theorem states that in an alternative algebra the subalgebra generated by any two elements is associative.

Alternative algebras are algebras satisfying the alternative property.

Alternative algebras are so named because they are precisely the algebras for which the associator is alternating.

Examples include all associative algebras, all alternative algebras, Jordan algebras, and the sedenions.

If A is associative and x + x, xx associate and commute with everything, then B is an alternative algebra.

In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative.

The most important examples of alternative algebras are the octonions (an algebra over the reals), and generalizations of the octonions over other fields.

Any alternative algebra may be made into a Malcev algebra by defining the Malcev product to be xy yx.

A corollary of Artin's theorem is that alternative algebras are power-associative, that is, the subalgebra generated by a single element is associative.

Although neither commutative nor associative, composition algebras have the special property of being alternative algebras, i.e. left and right multiplication preserves squares, a weakened version of associativity.

It retains an algebraic property called power associativity, meaning that if is a sedenion, , but loses the property of being an alternative algebra and hence cannot be a composition algebra.

Every associative algebra is obviously power-associative, but so are all other alternative algebras (like the octonions, which are non-associative) and even some non-alternative algebras like the sedenions and Okubo algebras.