The Real Subalgebras of $\mathfrak{so}_4(\mathbb{C})$ and $G_{2(2)}$

Classifying the subalgebras of a simple Lie algebra is a pursuit that stretches back to the work of Cartan on representations of simple Lie algebras. Mal'cev, in classifying orthogonal and symplectic representations of simple Lie algebras also found the semisimple subalgebras of $B_n, C_n,$ and $D_n$. Following that, Dynkin and Minchenko classified the semisimple subalgebras of the complex exceptional Lie algebras. We investigate the real subalgebras of a number of rank 2 Lie algebras: $\mathfrak{so}_4(\mathbb{C})$, its real forms, and the split real form of $G_2$, $G_{2(2)}$. In this thesis, we classify the real subalgebras of these Lie algebras up to inner automorphism (i.e., up to the adjoint action of the corresponding Lie group). For the matrix algebras above, we largely proceed with the help of copious amounts of linear algebra. For $G_{2(2)}$, we take advantage of the Cartan decomposition $G_{2(2)} = \mathfrak{k} \oplus \mathfrak{p}$, where $\mathfrak{k}$ is a compact subalgebra, to identify the semisimple and Levi-decomposable subalgebras of $G_{2(2)}$. To find the solvable subalgebras, we use the classifications of semisimple and nilpotent elements of $G_{2(2)}$, as well as our own classification of Jordan elements, to build nilpotent and solvable subalgebras.

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