  We study the analogue of Sidki's weak commutativity construction, defined
originally for groups, in the category of Lie algebras. This is the quotient
$\chi(\mathfrak{g})$ of the Lie algebra freely generated by two isomorphic
copies $\mathfrak{g}$ and $\mathfrak{g}^{\psi}$ of a fixed Lie algebra by the
ideal generated by the brackets $[x,x^{\psi}]$, for all $x$. We exhibit an
abelian ideal of $\chi(\mathfrak{g})$ whose associated quotient is a subdirect
sum in $\mathfrak{g} \oplus \mathfrak{g} \oplus \mathfrak{g}$ and we give
conditions for this ideal to be finite dimensional. We show that
$\chi(\mathfrak{g})$ has a subquotient that is isomorphic to the Schur
multiplier of $\mathfrak{g}$. We prove that $\chi(\mathfrak{g})$ is finitely
presentable or of homological type $FP_2$ if and only if $\mathfrak{g}$ has the
same property, but $\chi(\mathfrak{f})$ is not of type $FP_3$ if $\mathfrak{f}$
is a non-abelian free Lie algebra.
