Levi components of parabolic subalgebras of finitary Lie algebras

We characterize locally semisimple subalgebras l of sl(infinity), so(infinity), and sp(infinity) which are Levi components of parabolic subalgebras. Given I, we describe the parabolic subalgebras p such that l is a Levi component of p. We also prove that not every maximal locally semisimple subalgebra of a finitary Lie algebra is a Levi component. When the set of self-normalizing parabolic subalgebras p with fixed Levi component l is finite, we prove an estimate on its cardinality. We consider various examples which highlight the differences from the case of parabolic subalgebras of finite-dimensional simple Lie algebras.