  We give a complete classification, up to isometric isomorphism and scaling,
of $4$-dimensional metric Lie algebras $(\mathfrak{g},\langle \cdot,\cdot
\rangle)$ that admit a non-zero parallel skew-symmetric endomorphism. In
particular, we distinguish those metric Lie algebras that admit such an
endomorphism which is not a multiple of a complex structure, and for each of
them we obtain the de Rham decomposition of the associated simply connected Lie
group with the corresponding left invariant metric. On the other hand, we find
that the associated simply connected Lie group is irreducible as a Riemannian
manifold for those metric Lie algebras where each parallel skew-symmetric
endomorphism is a multiple of a complex structure.

Differential Geometry