Enveloping algebras of the nilpotent Malcev algebra of dimension five

Pérez-Izquierdo and Shestakov recently extended the PBW theorem to Malcev algebras. It follows from their construction that for any Malcev algebra M over a field of characteristic 6= 2, 3 there is a representation of the universal nonassociative enveloping algebra U(M) by linear operators on the polynomial algebra P (M). For the nilpotent non-Lie Malcev algebra M of dimension 5, we use this representation to determine explicit structure constants for U(M); from this it follows that U(M) is not power-associative. We obtain a finite set of generators for the alternator ideal I(M) ⊂ U(M) and derive structure constants for the universal alternative enveloping algebra A(M) = U(M)/I(M), a new infinite dimensional alternative algebra. We verify that the map ι : M→ A(M) is injective, and so M is special.