Malcev algebras play a role in the theory of
Moufang loops that generalizes the role of
Lie algebras in the theory of
groups. Namely, just as the tangent space of the identity element of a
Lie group forms a Lie algebra, the tangent space of the identity of a smooth Moufang loop forms a Malcev algebra. Moreover, just as a Lie group can be recovered from its Lie algebra under certain supplementary conditions, a smooth Moufang loop can be recovered from its Malcev algebra if certain supplementary conditions hold. For example, this is true for a connected, simply connected real-analytic Moufang loop.[1]
Any
alternative algebra may be made into a Malcev algebra by defining the Malcev product to be xy − yx.
The 7-sphere may be given the structure of a smooth Moufang loop by identifying it with the unit
octonions. The tangent space of the identity of this Moufang loop may be identified with the 7-dimensional space of imaginary octonions. The imaginary octonions form a Malcev algebra with the Malcev product xy − yx.