In analogy to the doubling of vector spaces introduced for the [L.sub.[infinity]] realization of Theorem 1 we will show that every nonassociative algebra has a realization as an [A.sub.[infinity]] algebra.
Let (V,*) be a nonassociative algebra and [V.sup.*] a vector space isomorphic to V with the isomorphism denoted by V [contains as member of] a [??] [a.sup.*] [member of] [V.sup.*].
This completes the proof that any nonassociative algebra can be embedded into an [A.sub.[infinity]] algebra.
By defining a new binary operation a-b = (1/2)(ab + ba) on an associative algebra over a field whose characteristic is not equal to 2, we obtain another important nonassociative algebra known as Jordan algebra.
Moreover, we introduce a new class of a nonassociative algebra called left almost algebra.
The Role of Nonassociative Algebra
in Projective Geometry
Maltsev used the name "Moufang-Lie algebras" for these nonassociative algebras
while Sagle  introduced the term "Malcev algebras." Equivalent defining identities of Maltsev algebras are pointed out in .
in Physics, Hadronic Press, Palm Harbor 1994; ActaAppl.
Nonlinear Elliptic Equations and Nonassociative Algebras
Nadirashvili, Tkachev, and Vladut present students, academics, and mathematicians with a collection of applications of noncommutative and nonassociative algebras, used to construct unusual solutions to nonlinear elliptic partial differential equations of the second order.
An Introduction to Nonassociative Algebras
. Dover Publications Inc., New York, 1995.
The original motivation to introduce the class of nonassociative algebras
known as Jordan algebras came from quantum mechanics (see [?]).