The subalgebras of the generalized special unitary algebra $\mathfrak{s}\mathfrak{u}(2,1)$

We classify the real subalgebras of the generalized special unitary algebra $\mathfrak{s}\mathfrak{u}(2,1)$, a non-compact real form of the complex special linear algebra ${\mathfrak{s}\mathfrak{l}}_3(\mathbb{C})$. Our approach combines Galois cohomology with the existing classification of complex subalgebras of ${\mathfrak{s}\mathfrak{l}}_3(\mathbb{C})$. This work completes the classification of real subalgebras of the non-compact real forms of ${\mathfrak{s}\mathfrak{l}}_3(\mathbb{C})$, since those of ${\mathfrak{s}\mathfrak{l}}_3(\mathbb{R})$ have already been classified.