Let $A$ and $B$ be subalgebras of a Lie algebra $L$ such that $B\subset N_L(A)$.
(a) Verify that the space $A+B$ is a subalgebra of $L$.
(b) Verify that $A\triangleleft(A+B)$ and $(A\cap B)\triangleleft B$.
(c) Prove that $(A+B)/A\cong B/(A\cap B)$.
My attempt:
(a) Since $A$ and $B$ are subalgebras $A+B$ is a subspace of $L$. We need to show that $[c_1,c_2]\in A+B$ for $c_1,c_2 \in A+B$. Now $[c_1,c_2]=[a_1+b_1,a_2+b_2]=[a_1,a_2]-[b_2,a_1]+[b_1,a_2]+[b_1,b_2]$ which is in $A+B$.
(b)Let $a_1,a_2 \in A$ and $b_1 \in B$ then $[a_1,a_2+b_1]=[a_1,a_2]-[b_1,a_2]$ which is in $A+B$. Hence, $A\triangleleft(A+B)$. Let $c \in A\cap B$ then $[c,b] \in A\cap B$, i.e. $(A\cap B)\triangleleft B$.
(c) I guess one should use the fact that if $I,J$ are ideals of $L$, there is a natural isomorphism between $(I +J)/J$ and $I/(I \cap J)$ but I cannot quite see how to apply it. Thanks in advance.
EDIT: If $B\triangleleft N_L(A)$ then (c) is trivial.