  When $\mathfrak h$ is a toral subalgebra of a Lie algebra $\mathfrak g$ over
a field $\mathbf k$, and $M$ a $\mathfrak g$-module on which $\mathfrak h$ also
acts torally, the Hochschild-Serre filtration of the Chevalley-Eilenberg
cochain complex admits a stronger form than for an arbitrary subalgebra. For a
semidirect product $\mathfrak g = \mathfrak h \ltimes \mathfrak k$ with
$\mathfrak h$ toral one has $H^*(\mathfrak g, M) \cong \bigwedge\mathfrak
h^{\vee} \bigotimes H^*(\mathfrak k,M)^{\mathfrak h} = H^*(\mathfrak h, \mathbf
k)\bigotimes H^*(\mathfrak k,M)^{\mathfrak h}$, and for a Lie poset algebra
$\mathfrak g$, that $H^*(\mathfrak g, \mathfrak g)$, which controls the
deformations of $\mathfrak g$, can be computed from the nerve of the underlying
poset. The deformation theory of Lie poset algebras, analogous to that of
complex analytic manifolds for which it is a small model, is illustrated by
examples.
