An Explicit Isomorphism Between the Three Dimensional Orthogonal Lie Algebra and the Special Linear Lie Algebra of Dimension $3$

Question

I know from various sources (http://homepages.warwick.ac.uk/~masdf/research/y4_fowlerwright.pdf and https://pi.math.cornell.edu/~hatcher/Other/Samelson-LieAlg.pdf are two) that the complex orthogonal Lie algebra of dimension 3, $\mathfrak{o}_3(\mathbb C)$, (by which I mean the space of skew-symmetric matrices of size 3) is isomporphic to $\mathfrak{sl}_2(\mathbb C)$, but I cannot come up with an explicit isomorphism. Can someone give me one?

Answer 1

For an explicit isomorphism of Lie algebras $$ \mathfrak{o}_3(\Bbb C)= \mathfrak{so}_3(\Bbb C)\cong \mathfrak{sl}_2(\Bbb C) $$ see here. One can find many more posts here on this site:

Is there an intelligent way to find the Lie algebra isomorphism $\mathfrak{sl}_2(\mathbb{C})\simeq\mathfrak{o}_3(\mathbb{C})$?

Lie algebra isomorphism between $\mathfrak{sl}(2,{\bf C})$ and $\mathfrak{so}(3,\Bbb C)$