Let $V$ be a completely reducible representation of a lie algebra $\mathfrak{g}$.
Show that $V^{\mathfrak{g}} = \{ v \in V : \rho_{x}(v) = 0\}$ is isomorphic to $V_{\mathfrak{g}} = V / \mathfrak{g}V$, where $\mathfrak{g}V = \{ \rho_{x}(v) : x \in \mathfrak{g}, v \in V\}$
We have $V = \oplus_{i \in I} V_{i}$ where $V_{i}$ are irreducible. Do I have that $V = V^{\mathfrak{g}} \oplus (V^{\mathfrak{g}})^{c}$, and then would $V^{\mathfrak{g}} \hookrightarrow V \xrightarrow{\pi} V_{\mathfrak{g}}$ be an isomorphism? If so, where is the assumption of complete reducibility used?
For an irreducible representation $W$ we have two cases:
$$ acts trivially on $W$, in which case $W^ = W$ and $ W = 0$.
$$ acts non-trivially on $W$. In this case, $W^$ is a proper subrepresentation of $W$ and $ W$ is a non-zero subrepresentation of $W$. Consequently, $W^ = 0$ and $ W = W$ because $W$ is irreducible.
We also observe that for every representation $W$ and decomposition $W = \bigoplus_{i ∈ I} W_i$ into subrepresentations we have $W^ = \bigoplus_{i ∈ I} W_i^$ and $ W = \bigoplus_{i ∈ I} W_i$.
Let now $V = ⨁_{i ∈ I} V_i$ be a decomposition into irreducible subrepresentations. Let $J ⊆ I$ be the subset of all those indices $j$ for which $$ acts trivially on $V_j$. We find that $$ V^ = \bigoplus_{i ∈ I} V_i^ = \bigoplus_{j ∈ J} V_j $$ and that $$ V = \bigoplus_{i ∈ I} V_i = \bigoplus_{i ∈ I \setminus J} V_i \,. $$ It follows that $$ V / V = \bigoplus_{i ∈ I} V_i \Bigg/\!\! \bigoplus_{i ∈ I \setminus J} V_i ≅ \bigoplus_{j ∈ J} V_j = V^ \,. $$