A Lie algebra over a field is said to be finitary if it is isomorphic to a subalgebra of the Lie algebra of all finite rank operators on a vector space. An inner ideal of a Lie algebra \(L\) over a field is a subspace \(I\) satisfying \([I\,[I,L]]\subseteq I\). Central finitary simple Lie algebras over characteristic zero fields has been classified by \textit{A. A. Baranov} [J. Algebra 219, No. 1, 299--329 (1999; Zbl 0949.17008)]. In the paper under review, the inner ideal structure of finitary special linear algebras and finitary Lie algebras of skew-operators is described. As a consequence of the structure inner ideal results obtained and Baranov's classification, the authors prove that any finitary central simple Lie algebra over a field of characteristic zero satisfies the descending chain condition on principal inner ideals.