The Subalgebras of the Real Forms of \(\mathfrak{sl}_3(\mathbb{C})\)

We classify the subalgebras of the real forms the complex linear algebra $\mathfrak{sl}_3(\mathbb{C})$, namely the real special linear algebra $\mathfrak{sl}_3(\mathbb{R})$, the special unitary algebra $\mathfrak{su}(3)$, and the generalized special unitary algebra $\mathfrak{su}(2,1)$. Our approach applies Galois cohomology to the known classification of complex subalgebras of $\mathfrak{sl}_3(\mathbb{C})$. The subalgebras of $\mathfrak{sl}_3(\mathbb{R})$ were previously classified by Winternitz using different techniques. We recover this classification using our cohomological approach and amend minor inaccuracies. Our work, however, constitutes the first complete classifications of the subalgebras of $\mathfrak{su}(3)$ and $\mathfrak{su}(2,1)$. In addition to illuminating the internal structure of the real forms of $\mathfrak{sl}_3(\mathbb{C})$, our methodology provides a pathway for future investigations into the subalgebra structure of higher-dimensional cases. In addition, the present work and its extensions offer potential applications in representation theory, applied mathematics, and physics.