for all x and y in the algebra.
Both of these identities together imply that the associator is totally skew-symmetric. That is,
for any permutation σ. It follows that
The associator of an alternative algebra is therefore alternating. Conversely, any algebra whose associator is alternating is clearly alternative. By symmetry, any algebra which satisfies any two of:
- left alternative identity:
- right alternative identity:
- flexible identity:
is alternative and therefore satisfies all three identities.
An alternating associator is always totally skew-symmetric. The converse holds so long as the characteristic of the base field is not 2.
- Every associative algebra is alternative.
- The octonions form a non-associative alternative algebra, a normed division algebra of dimension 8 over the real numbers.
- More generally, any octonion algebra is alternative.
- The sedenions and all higher Cayley–Dickson algebras lose alternativity.
- All antiassociative algebras are by definition not alternative.
Artin's theorem states that in an alternative algebra the subalgebra generated by any two elements is associative. Conversely, any algebra for which this is true is clearly alternative. It follows that expressions involving only two variables can be written unambiguously without parentheses in an alternative algebra. A generalization of Artin's theorem states that whenever three elements in an alternative algebra associate (i.e., ), the subalgebra generated by those elements is associative.
A corollary of Artin's theorem is that alternative algebras are power-associative, that is, the subalgebra generated by a single element is associative. The converse need not hold: the sedenions are power-associative but not alternative.
hold in any alternative algebra.
In a unital alternative algebra, multiplicative inverses are unique whenever they exist. Moreover, for any invertible element and all one has
This is equivalent to saying the associator vanishes for all such and . If and are invertible then is also invertible with inverse . The set of all invertible elements is therefore closed under multiplication and forms a Moufang loop. This loop of units in an alternative ring or algebra is analogous to the group of units in an associative ring or algebra.
The projective plane over any alternative division ring is a Moufang plane.
The close relationship of alternative algebras and composition algebras was given by Guy Roos in 2008: He shows (page 162) the relation for an algebra A with unit element e and an involutive anti-automorphism such that a + a* and aa* are on the line spanned by e for all a in A. Use the notation n(a) = aa*. Then if n is a non-singular mapping into the field of A, and A is alternative, then (A,n) is a composition algebra.
- Schafer (1995) p.27
- Schafer (1995) p.28
- Conway, John Horton; Smith, Derek A. (2003). On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. A. K. Peters. ISBN 1-56881-134-9. Zbl 1098.17001.
- Schafer (1995) p.29
- Schafer (1995) p.30
- Zhevlakov, Slin'ko, Shestakov, Shirshov. (1982) p.151
- Zhevlakov, Slin'ko, Shestakov, Shirshov. (1982)
- Guy Roos (2008) "Exceptional symmetric domains", §1: Cayley algebras, in Symmetries in Complex Analysis by Bruce Gilligan & Guy Roos, volume 468 of Contemporary Mathematics, American Mathematical Society
- Schafer, Richard D. (1995). An Introduction to Nonassociative Algebras. New York: Dover Publications. ISBN 0-486-68813-5. Zbl 0145.25601.
- Zhevlakov, K.A.; Slin'ko, A.M.; Shestakov, I.P.; Shirshov, A.I. (1982) . Rings that are nearly associative. Academic Press. ISBN 0-12-779850-1. MR 0518614. Zbl 0487.17001.