  We explore general intrinsic and extrinsic conditions that allow the
transitivity of the relation of being a Lie ideal, in the sense that if a Lie
algebra $\mathfrak{h}$ is a subideal of a Lie algebra $\mathfrak{g}$ (i.e.
there exist Lie subalgebras
$\mathfrak{l}_0,\mathfrak{l}_1,\dots,\mathfrak{l}_n$ of $\mathfrak{g}$ with
$\mathfrak{h}=\mathfrak{l}_0\unlhd \mathfrak{l}_1 \unlhd\cdots \unlhd
\mathfrak{l}_n=\mathfrak{g}$), then $\mathfrak{h}$ is an ideal of
$\mathfrak{g}$. We also prove that perfect Lie algebras of arbitrary dimension
and over any field are intrinsically characterized by transitivity of this
type; In particular, we show that a Lie algebra $\mathfrak{h}$ is perfect (i.e.
$\mathfrak{h}=[\mathfrak{h}, \mathfrak{h}]$) if and only if for any Lie algebra
$\mathfrak{g}$ such that $\mathfrak{h}$ is a subideal of $\mathfrak{g}$, it
follows that $\mathfrak{h}$ is an ideal of $\mathfrak{g}$.
