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A subalgebra $H$ of a finite dimensional Lie algebra $L$ is said to be a $\mathrm{SCAP}$-subalgebra if there is a chief series $0=L_0\subset L_1\subset ...\subset L_t=L$ of $L$ such that for every $i=1,2,...,t$, 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 $\mathrm{SCAP}$-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 $\mathrm{SCAP}$-subalgebras and give some sufficient conditions for a Lie algebra to be solvable or supersolvable.