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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
299:. 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
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The 7-sphere may be given the structure of a smooth
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230:{\displaystyle (xy)(xz)=((xy)z)x+((yz)x)x+((zx)x)y.}
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414:Mal'cev, A. I. (1955), "Analytic loops",
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318:Malcev-admissible algebra
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454:-related article is a
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96:{\displaystyle xy=-yx}
63:nonassociative algebra
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277:is a Malcev algebra.
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282:alternative algebra
18:Moufang–Lie algebra
398:"Mal'tsev algebra"
346:Seminar Sophus Lie
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352:: 65–68.
297:octonions
261:Lie group
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312:See also
268:Examples
244:(1955).
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