  We study compatible actions (introduced by Brown and Loday in their work on
the non-abelian tensor product of groups) in the category of Lie algebras over
a fixed ring. We describe the Peiffer product via a new diagrammatic approach,
which specializes to the known definitions both in the case of groups and of
Lie algebras. We then use this approach to transfer a result linking compatible
actions and pairs of crossed modules over a common base object $L$ from groups
to Lie algebras. Finally, we show that the Peiffer product, naturally endowed
with a crossed module structure, has the universal property of the coproduct in
$\mathbf{XMod}_L(\mathbf{Lie}_R)$.
