An envelope for Malcev algebras

We prove that for every Malcev algebra M there exist an algebra U(M) and a monomorphism ι :M→U(M)− of M into the commutator algebra U(M)− such that the image of M lies into the alternative center of U(M), and U(M) is a universal object with respect to such homomorphisms. The algebra U(M), in general, is not alternative, but it has a basis of Poincaré–Birkhoff–Witt type over M and inherits some good properties of universal enveloping algebras of Lie algebras. In particular, the elements of M can be characterized as the primitive elements of the algebra U(M) with respect to the diagonal homomorphism Δ :U(M)→U(M) U(M). An extension of Ado–Iwasawa theorem to Malcev algebras is also proved.