I previously determined that the Lie algebra of $O(3,\mathbb C)$ is the set of skew symmetric matrices and that the Lie algebra of $SL_2(\mathbb C)$ is the set of traceless matrices. I am now trying to show that the two Lie algebras are isomorphic. But the calculations blew up and I'm now not so sure anymore if I'm on the right track.
Here is my idea:
Let $\mathfrak o$ denote the Lie algebra of $O(3,\mathbb C)$ and $\mathfrak g$ denote the Lie algebra of $SL_2$.
Define a map $\varphi : \mathfrak g \to \mathfrak o$ by defining $$ g = (p,q) \mapsto (P,Q,P\times Q) $$ where $p,q$ are the columns of $g$ and $P = (p,0), Q=(q,0)$. (that's padding with one $0$)
First, I am trying to show that it's in fact ahomomorphism:
$$ \begin{align} \varphi(p+p', q+q') &= (P+P', Q+Q', (P+P')\times (Q+Q')) \\ &= (P, Q+Q', (P+P')\times (Q+Q')) + (P', Q+Q', (P+P')\times (Q+Q')) \\ &= (P, Q, (P+P')\times (Q+Q')) + (P, Q', (P+P')\times (Q+Q')) + (P', Q, (P+P')\times (Q+Q')) + (P', Q', (P+P')\times (Q+Q')) \\ &= (P, Q, P\times (Q+Q')) +(P, Q, P' \times (Q+Q')) + (P, Q', P\times (Q+Q')) \\ &+ (P, Q', P'\times (Q+Q')) + (P', Q, P\times (Q+Q'))+(P', Q, P' \times (Q+Q')) \\ &+ (P', Q', P\times (Q+Q')) + (P', Q', P'\times (Q+Q')) \end{align}$$
But this is only getting bigger when in fact it should eventually become a sum of just 2 terms.
Does my idea work or is it wrong?