Some properties of Camina and $n$-Baer Lie algebras

Let $I$ be a non-zero proper ideal of a Lie algebra $L$. Then $(L, I)$ is called a Camina pair if $I \subseteq [x,L]$, for all $x \in L\setminus I$. Also, $L$ is called a Camina Lie algebra if $(L, L^2)$ is a Camina pair. We first give some properties of Camina Lie algebras, and then show that all Camina Lie algebras are soluble. Also, a new notion of $n$-Baer Lie algebras is introduced, and we investigate some of its properties, for $n=1, 2$. A Lie algebra $L$ is said to be $2$-Baer if for any one dimensional subalgebra $K$ of $L$, there exists an ideal $I$ of $L$ such that $K$ is an ideal of $I$. Finally, we study three classes of Lie algebras with $2$-subideal subalgebras and give some relations among them.