The subalgebras of $\mathfrak{s}\mathfrak{o}(4,\mathbb{C})$

We classify the solvable subalgebras, semisimple subalgebras, and Levi decomposable subalgebras of $\mathfrak{s}\mathfrak{o}(4,\mathbb{C})$, up to inner automorphism. By Levi’s Theorem, this is a full classification of the subalgebras of $\mathfrak{s}\mathfrak{o}(4,\mathbb{C})$.