SCAP-subalgebras of Lie algebras

summary:A subalgebra $H$ of a finite dimensional Lie algebra $L$ is said to be a $\rm SCAP$-subalgebra if there is a chief series $0=L_0\subset L_1\subset \ldots \subset L_t=L$ of $L$ such that for every $i=1,2,\ldots ,t$, we have $H+L_i=H+L_{i-1}$ or $H\cap L_i=H\cap L_{i-1}$. This is analogous to the concept of $\rm SCAP$-subgroup, which has been studied by a number of authors. In this article, we investigate the connection between the structure of a Lie algebra and its $\rm SCAP$-subalgebras and give some sufficient conditions for a Lie algebra to be solvable or supersolvable