I am asked to prove that the Lie algebras $\mathfrak{sl}_2(\mathbb C), \; \mathfrak{so}_3(\mathbb C)$ are Lie isomorphic to each other.
We have the standard basis for $\mathfrak{sl}_2(\mathbb C)$ to be $\{e = \left(\begin{matrix}0 & 1 \\ 0 & 0\end{matrix}\right), \; f = \left(\begin{matrix}0 & 0 \\ 1 & 0\end{matrix}\right), \; h = \left(\begin{matrix}1 & 0 \\ 0 & -1\end{matrix}\right)\}$
which satisfy the bracket relations $[e,f] = h, \; [h,e] = 2e,\; [h,f] = - 2f$
After a little work, I found the basis of $\mathfrak{so}_3(\mathbb C)$ to be: $\{a = \left(\begin{matrix}0&1&0 \\ 0&0&0 \\ -1&0&0 \end{matrix}\right), \; b = \left(\begin{matrix}0&0&1 \\ -1&0&0 \\ 0&0&0 \end{matrix}\right), \; c = \left(\begin{matrix}0&0&0 \\ 0&1&0 \\ 0&0&-1 \end{matrix}\right) \}$
which I found to have the bracket relations: $[a, b] = c , \; [c, a] = -a, \; [c , b] = b$
I'm now stuck, because I don't know to map the basis $\{e, f, h\}$ to a basis expressed in terms of $\{a, b, c\}$ such that the bracket relations are maintained.
I would appreciate it if someone could highlight how I should augment the basis of $\mathfrak{so}_3(\mathbb C)$ so that I can construct the Lie isomorphism, as well as explain what the general method for finding it might be for similar situations.
Thank you in advance.