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 a Lie ideal, in the sense that if a Lie algebra h is a subideal of a Lie algebra g (i. e. there exist Lie subalgebras l₀, l₁, , lₙ of g with h=l₀ l₁ lₙ=g), then h is an ideal of 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 h is perfect (i. e. h=h, h) if and only if for any Lie algebra g such that h is a subideal of g, it follows that h is an ideal of g.