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Perez-Izquierdo and Shestakov recently extended the PBW theorem to Malcevalgebras. It follows from their construction that for any Malcev algebra $M$over a field of characteristic $\ne 2, 3$ there is a representation of theuniversal nonassociative enveloping algebra $U(M)$ by linear operators on thepolynomial algebra $P(M)$. For the nilpotent non-Lie Malcev algebra$\mathbb{M}$ of dimension 5, we use this representation to determine explicitstructure constants for $U(\mathbb{M})$; from this it follows that$U(\mathbb{M})$ is not power-associative. We obtain a finite set of generatorsfor the alternator ideal $I(\mathbb{M}) \subset U(\mathbb{M})$ and derivestructure constants for the universal alternative enveloping algebra$A(\mathbb{M}) = U(\mathbb{M})/I(\mathbb{M})$, a new infinite dimensionalalternative algebra. We verify that the map $\iota\colon \mathbb{M} \toA(\mathbb{M})$ is injective, and so $\mathbb{M}$ is special.

Enveloping algebras of the nilpotent Malcev algebra of dimension five