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 $\mathfrak{ g}$.
Categories: math.RA