The Cartan subalgebra of $\mathfrak{sl}_2(\mathbb{C})$ is $$\mathfrak{h} = \mathbb{C} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$ which is $1$-dimensional and hence can be identified with $\mathbb{C}$.
Now, given real numbers $\alpha,\beta \in \mathbb{R}$, how can one determine the value of $\langle \alpha,\beta \rangle$ where $\langle \cdot,\cdot \rangle$ is the Killing form?
I imagine the calculation would involve using the killing form to transfer $\mathfrak{h} \to \mathfrak{h}^*$. Is there a quick and easy way to do this?
With $\alpha$ and $\beta$ defined as $\alpha=\alpha \bigl( \begin{smallmatrix} 1&0\\0&-1\end{smallmatrix} \bigr)$, $\beta=\beta \bigl( \begin{smallmatrix} 1&0\\0&-1\end{smallmatrix} \bigr)$, and taking $x=\bigl(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\bigr)$, $h=\bigl(\begin{smallmatrix}1&0\\0&-1\end{smallmatrix}\bigr)$, $y=\bigl(\begin{smallmatrix}0&0\\1&0\end{smallmatrix}\bigr)$ as a basis for $\mathfrak{sl}_2(\mathbb{C})$, we have:
$$ ad_\alpha = \left(\begin{matrix} 2\alpha & 0 & 0\\ 0&0&0 \\ 0&0&-2\alpha \end{matrix} \right), ad_\beta=\left(\begin{matrix} 2\beta & 0 & 0\\ 0&0&0 \\ 0&0&-2\beta \end{matrix} \right) $$ The usual definition for the Killing form is $\kappa(x,y)=Tr(ad_x \cdot ad_y)$. Using this we get $\kappa(\alpha,\beta)=Tr(ad_\alpha \cdot ad_\beta)=8\alpha\beta$.