Serial subalgebras of finitary Lie algebras

A Lie subalgebra $L$ of ${\mathfrak{gl}_{{\mathbb{K}}}(V)}$ is said to be finitary if it consists of elements of finite rank. We show that, if $L$ acts irreducibly on $V$, and if $V$ is infinite-dimensional, then every non-trivial ascendant Lie subalgebra of $L$ acts irreducibly on $V$ too. When $\operatorname{Char} \mathbb{K}\neq 2$, it follows that the locally solvable radical of such $L$ is trivial. In general, locally solvable finitary Lie algebras over fields of characteristic $\neq 2$ are hyperabelian.