Show $\phi$ is a isomorphism as a lie algebra homomorphism
$\phi: \textbf{su}_2 \bigotimes_{\mathbb{R}} \mathbb{C}\rightarrow sl_2(\mathbb{C})$ and $\phi: a(I \bigotimes 1)+b(J \bigotimes 1)+c(K \bigotimes 1) \rightarrow (ai+b)X+(ai-b)Y+ciH$
Where {$I, J, K$} is a basis of $\textbf{su}_2$ and $a, b, c \in \mathbb{C}$ and {$X, Y, H$} is a basis of $\textbf{sl}_2\mathbb{C}$
$\textbf{What I know:}$ Requirements for $\phi$ to be a lie-algebra isomorphism:
- lie-bracket must be preserved $\phi([X,Y])=[\phi(X), \phi(Y)]$
- $\phi$ must be 1:1 and onto
- I believe that since $\phi$ is a lie algebra homomorphism, then we must just show $\phi$ is a vector-space isomorphism
My problem is I can't apply these facts to the given map; I just can't get my head around it. Any thanks would be very much appreciated