I'm sure I'm missing something really obvious here. This seems too stupid.
On page 47 of Erdmann & Wildon's Introduction to Lie Algebras, we have the following set up. Let $L$ be a Lie subalgebra of $\mathfrak{gl}(V)$ of dimension $\geq 1$, and let $A \leq L$ be a maximal Lie subalgebra and let $\overline{L} = L/A$. Then they define $\varphi: A \to \mathfrak{gl}(\overline{L})$ by $\varphi(a)(x+A) := [a,x] + A$.
My question is: why do we not have $\varphi = 0$? Or do we have this? I mean, if $a \in A, x+A \in \overline{L}$, then $\varphi(a)(x+A) = [a,x] + A = [a+A,x+A]=[0,x+A]=0$, right?
Maybe I'm not understanding how the braket is defined on the quotient, or something.
Let's have an example. Let $L = span_{\mathbb F} \{x, y\}$, and $[x, y] = y$ (Note that every two dimensional Lie algebra are of this form). Let $A = span\{x\}$. Then the map $$\phi: A \to gl(\bar L)$$
is $\phi(cx)(y + A) = [cx, y] + A = cy+ A$. This is not a zero map.