About the definition of $\mathrm{G}_2$

Question

What is the actual definition of the exceptional Lie group $\mathrm{G}_2$? In the literature I always find things like 'It can be seen as the automorphism group of the algebra of octonions' or 'Given a 7-dimensional vector space $V$ over the real, it is isomorphic to the subgroup of $\mathrm{GL}(V)$ fixing a particular $3$-form'.

I guess the answer lies in the theory of representations of Lie groups/Lie algebras, but I am afraid there is a long path to take before getting to the exceptional Lie groups. Any suggestion for approaching this subject would be appreciated.

Answer 1

There is no such thing as the Lie group $G_2$. There is the exceptional complex Lie algebra $\mathfrak{g}_2$, which is a $14$-dimensional simple Lie algebra over $\mathbb C$. And there are several connected real lie groups $G$ such that the complexification of the corresponding Lie algebra is (isomorphic to) $\mathfrak{g}_2$. One of them (and, up to isomorphism, the only one which is compact) is indeed the group of automorphisms of the octonions. This was proved by Elie Cartan in 1914.

Answer 2

What do you mean definition? Simple Lie algebras are classified by the root system of the adjoint action. These, in turn, are classified by Dynkin diagrams.

The root system for $\mathfrak{g}_2$ corresponds to one of these diagrams. One can reconstruct the Lie algebra $\mathfrak{g}_2$ (up to isomorphism) from the root system. As pointed out by Jose Carlos Santos, there are several connected Lie groups which correspond to this Lie algebra (one of which is the automorphisms of the octonians).

If you want something more concrete to think of when visualizing $G_2$, I'd like to point out that John Baez and John Huerta have a fantastic way to visualize this group (the automorphisms of the octonians) as the symmetry group of one ball rolling on another one, when the ratios of the radii of the two balls is 1:3. You can find that here https://arxiv.org/pdf/1205.2447.pdf