Locally semisimple and maximal subalgebras of the finitary Lie algebras   $gl(\infty)$, $sl(\infty)$, $so(\infty)$, and $sp(\infty)$
  We describe all locally semisimple subalgebras and all maximal subalgebras of the finitary Lie algebras $\gl(\infty), \sl(\infty), \so(\infty)$, and $\sp(\infty)$. For simple finite--dimensional Lie algebras these classes of subalgebras have been described in the classical works of A. Malcev and E. Dynkin.   Key words (2000 MSC): 17B05 and 17B65. 
Ivan Dimitrov, Ivan Penkov