On the transitivity of Lie ideals and a characterization of perfect Lie algebras

We explore general intrinsic and extrinsic conditions that allow the transitivity of the relation of being an ideal in Lie algebras. 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 \(\mathfrak {h}\) $\mathfrak{h}$ is perfect (i.e., \(\mathfrak {h}=[\mathfrak {h}, \mathfrak {h}]\) $\mathfrak{h}=[\mathfrak{h},\mathfrak{h}]$ ) if and only if for all Lie algebras \(\mathfrak {k}, \mathfrak {g}\) $\mathfrak{k},\mathfrak{g}$ such that \(\mathfrak {h}\) $\mathfrak{h}$ is an ideal of \(\mathfrak {k}\) $\mathfrak{k}$ and \(\mathfrak {k}\) $\mathfrak{k}$ is an ideal of \(\mathfrak {g}\) $\mathfrak{g}$ , it follows that \(\mathfrak {h}\) $\mathfrak{h}$ is an ideal of \(\mathfrak {g}\) $\mathfrak{g}$ .