  Perez-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 $\ne 2, 3$ 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
$\mathbb{M}$ of dimension 5, we use this representation to determine explicit
structure constants for $U(\mathbb{M})$; from this it follows that
$U(\mathbb{M})$ is not power-associative. We obtain a finite set of generators
for the alternator ideal $I(\mathbb{M}) \subset U(\mathbb{M})$ and derive
structure constants for the universal alternative enveloping algebra
$A(\mathbb{M}) = U(\mathbb{M})/I(\mathbb{M})$, a new infinite dimensional
alternative algebra. We verify that the map $\iota\colon \mathbb{M} \to
A(\mathbb{M})$ is injective, and so $\mathbb{M}$ is special.
