= Solution
Let $V=\mathbb C^4$. The <Special linear Lie algebra> $\mathfrak{sl}(V)$ acts on $\Lambda^2V$. The wedge product
$$
\Lambda^2V\times\Lambda^2V\longrightarrow\Lambda^4V\cong\mathbb C
$$
is a nondegenerate symmetric bilinear form, and the action preserves it because $\mathfrak{sl}(V)$ acts trivially on $\Lambda^4V$. This gives an injective homomorphism
$$
\mathfrak{sl}_4\longrightarrow\mathfrak{so}(\Lambda^2V)\cong\mathfrak{so}_6.
$$
Both Lie algebras have dimension $15$, so the map is an isomorphism. This realizes the <Isomorphism between so6 and sl4>.
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