Page "Alternative algebra" Paragraph 21

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 without parenthesis unambiguously 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.