Let $k$ be an algebraically closed characteristic zero field, consider the $\mathfrak{gl}_{1}$ representation
$$ \rho: \mathfrak{gl}_{1} \rightarrow k^{2} $$
given by
$$ \rho(x) = \begin{pmatrix} \alpha & x \\\ 0 & \beta \end{pmatrix} $$
Then this has a subrepresentation
$$ \\{ \begin{pmatrix} a \\\ 0 \end{pmatrix} : a \in k \\} $$
where $\rho(x)$ just acts as a scalar $\alpha$, so this subrep is isomorphic to $k_{\alpha}$
The quotient representation also just acts as a scalar $\beta$.
So $\alpha, \beta$ are the weights of $k^{2}$ (since $k_{\alpha}, k_{\beta}$ are the composition factors of $k^{2}$.)
I know that a representation of a nilpotent Lie algebra decomposes as the direct sum of its weight spaces. (Also, the weight spaces are equal to the generalised eigenspaces of $\rho(x)$ for a generic element $x \in \mathfrak{gl}_{1}$.)
$\mathfrak{gl}_{1}$ is nilpotent, so for $\alpha \ne \beta$, I am trying to decompose $k^{2}$ as
$$ k^{2} = k_{\alpha} \oplus k_{\beta} $$
But I cannot see what $k_{\beta}$ should be as a subrepresentation of $k^{2}$?
Any help would be appreciated!