I'm going through a proof of the theorem, any finite dimensional module of a semisimple Lie algebra is semisimple, from page 10 of these notes (pdf). I am having a difficult time understanding few parts of it and I would appreciate any explanations. I labeled the parts I would like to have some explanations (1), (2), (3) in the proof below. Thank you very much.

The proof goes as follows: Let $L$ be a semisimple Lie algebra where the base field $K$. It suffices to show that any submodule $A$ of a finite dimensional $L$-module $V$ has a complement which is invariant under $L$. First the case when $A$ is of codimension $1$ is proved (which I am fine with).

The general case (which I paraphrase the proof): We are looking for a projection $\pi: V \rightarrow A$, which commutes with the action of $L$. If we can find such a projection, then we are done because $V = A \oplus \ker(\pi)$. Let $\mathcal{V}$ and $\mathcal{A}$ denote the spaces of linear maps $V \rightarrow A$ whose restriction to $A$ is a homothety respectively $0$. $\mathcal{A}$ is a submodule of codimension $1$ in $\mathcal{V}$ (1).

Any element of $\mathcal{V}$ with non-zero restriction to $A$ is a scalar multiple of a projection from $V$ onto $A$ (2). The space $\mathcal{V}$ is mapped into $\mathcal{A}$ under the action of $L$ on $Hom(V,A)$, since
$$
(x \cdot \phi)(a) = x \cdot \phi(a)- \phi(x \cdot a) = 0
$$

for any $a \in A$ and any $\phi$ which is a homothery on $A$. The last identity also shows that those $\phi \in \mathcal{V}$ such that $K \phi$ is stable under $L$ and their restriction to $A$ are not zero are exactly the multiples of the projections $V \rightarrow A$ which commutes with $L$ (3). Hence, by our assumption we can find a submodule $K \phi$ such that $\mathcal{V} = \mathcal{A} \oplus K \phi$. Since the restriction of $\phi$ to non-zeoro we are done.