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for all ''x'' and ''y'' in ''A''. The quadratic form ''N'' is often referred to as a (''square'') ''norm'' on ''A'', although it is not a [[norm (mathematics)|norm]] in the usual sense. Composition algebras are also called '''normed algebras''' (not to be confused with [[normed algebra]]s in the sense of functional analysis).
==Structure theorem==
Every composition algebra over a field ''K'' can be obtained by repeated application of the [[Cayley-Dickson construction]] starting from ''K'' (if the [[characteristic (algebra)|characteristic]] of ''K'' is different from 2) or a 2-dimensional composition subalgebra (if char(''K'') = 2). The possible dimensions of a composition algebra are 1, 2, 4, and 8.
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