Observations about the Lie algebra ${\mathfrak{g}}_2\subset \mathfrak{s}\mathfrak{o}(7)$, associative $3$-planes, and $\mathfrak{s}\mathfrak{o}(4)$ subalgebras

We make several observations relating the Lie algebra ${\mathfrak{g}}_2\subset \mathfrak{s}\mathfrak{o}(7)$, associative $3$-planes, and $\mathfrak{s}\mathfrak{o}(4)$ subalgebras. Some are likely well-known but not easy to find in the literature, while other results are new. We show that an element $X\in {\mathfrak{g}}_2$ cannot have rank $2$, and if it has rank $4$ then its kernel is an associative subspace. We prove a canonical form theorem for elements of ${\mathfrak{g}}_2$. Given an associative $3$-plane $P$ in ${\mathbb{R}}^7$, we construct a Lie subalgebra $\mathrm{\Theta }(P)$ of $\mathfrak{s}\mathfrak{o}(7)={\mathrm{\Lambda }}^2({\mathbb{R}}^7)$ that is isomorphic to $\mathfrak{s}\mathfrak{o}(4)$. This $\mathfrak{s}\mathfrak{o}(4)$ subalgebra differs from other known constructions of $\mathfrak{s}\mathfrak{o}(4)$ subalgebras of $\mathfrak{s}\mathfrak{o}(7)$ determined by an associative $3$-plane. These are results of an NSERC undergraduate research project. The paper is written so as to be accessible to a wide audience.