On the theta completions and strong theta completions for proper subalgebras

For a proper subalgebra H of a Lie algebra L, we say a subalgebra C of L is a -completion for H if L = H;C while HL C and if C1=HL is a proper subalgebra of C=HL such that C1=HL C L=HL, then L , H + C1. A -completion C of H is called s- -completion, if C = L or there exists a subalgebra B of L such that C l B and B is not a -completion for H. By these notions we give some characterisations for solvability of Lie algebras.