Observations about the Lie algebra g2⊂so(7), associative 3-planes, and so(4) subalgebras

We make several observations relating the Lie algebra $g_2\subset \mathrm{so}\left(7\right)$, associative 3-planes, and $\mathrm{so}\left(4\right)$ subalgebras. Some are likely well-known but not easy to find in the literature, while other results are new. We show that an element $X\in g_2$ 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 $g_2$. Given an associative 3-plane $P$ in $R^7$, we construct a Lie subalgebra $\Theta \left(P\right)$ of $\mathrm{so}\left(7\right)={\Lambda }^2\left(R^7\right)$ that is isomorphic to $\mathrm{so}\left(4\right)$. This $\mathrm{so}\left(4\right)$ subalgebra differs from other known constructions of $\mathrm{so}\left(4\right)$ subalgebras of $\mathrm{so}\left(7\right)$ 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.