First of all, my question is written in $II)$ section.
I) Groups and Representations
In group theory a representation is a map that is defined as:
$$\rho: G \to GL(V) \tag{1}.$$
You take a element from $G$ and creates a object in $GL(V)$ that acts on a vector space $V$.
Using the pushfoward $\cdot_{*}$ it is possible to create representations for the lie algebras as well:
$$\rho_{*}: \mathfrak{g} \to \mathrm{End}(V) \tag{2}.$$
Furthermore, with the exponential map, $\mathrm{exp}$, you can "translate" the technology of standard Lie group theory (since in $(1)$ we are dealing with groups), into lie algebra representations as:
$\require{AMScd}$ \begin{CD} \mathfrak{g}@>{\rho_{*}}>> \hspace{0.4cm}\mathrm{End}(V)\\ @V{\mathrm{exp}}VV @VV{\mathrm{exp}}V\\ G @>{\rho}>> GL(V) \end{CD}
Now, a representation is said to be trivial if:
$\require{AMScd} \tag{3}$ \begin{CD} \rho:G\to GL(V) \\ \hspace{1.5cm}a \mapsto \rho(a) := \mathrm{Id}_{GL(V)}, \end{CD}
and
$\require{AMScd} \tag{4}$ \begin{CD} \rho_{*}:\mathfrak{g}\to \mathrm{End}(V) \\ \hspace{2cm}a_{\mathfrak{g}} \mapsto \rho_{*}(a_{\mathfrak{g}}) := 0_{\mathrm{End}(V)}. \end{CD}
II) My Question
My question is: considering the explicit example of the lie algebra of $G=SU(2)$, the $\mathfrak{g}=\mathfrak{su}(2)$. Why when do we choose the representation of the lie algebra as: $\rho_{*}:\mathfrak{su}(2)\to \mathrm{End}(\mathbb{C})$ we get:
$$\rho_{*}(a_{\mathfrak{su}(2)}) = 0_{\mathrm{End}(\mathbb{C})} \tag{5}$$
and when do we choose the representation of the lie algebra as: $\rho_{*}:\mathfrak{su}(2)\to \mathrm{End}(\mathbb{C}^2)$ we get:
$$\rho_{*}(a_{\mathfrak{su}(2)}) \neq 0_{\mathrm{End}(\mathbb{C}^2)} ?\tag{6}$$
Another two ways to ask the same question:
$1)$ Why do we get the trivial representation of $SU(2)$, and its lie algebra, when we choose $V=\mathbb{C}$?
$2)$ Why do we not get the trivial representation of $SU(2)$, and its lie algebra, when we choose $V=\mathbb{C}^2$?