Bilinear forms in compact/semisimple Lie group theory

(Edit: I rewrote this answer. In the first draft I tried to take some shortcuts and found that they didn't work.)

Let $G$ be a compact Lie group acting faithfully on a f.d. vector space $V$ over $\mathbb{C}$. It's a nice exercise to show that every f.d. irreducible representation of $G$ appears in some tensor product $V^{\otimes n} \otimes (V^*)^{\otimes m}$ (see, for example, this old MO question). What this implies is that the entire structure of the category $\operatorname{Rep}_\text f(G)$ of f.d. representations of $G$ is contained in the data of the invariant tensors $\operatorname{Hom}(V^{\otimes m}, V^{\otimes n})^G$: more formally, these invariant tensors describe the subcategory of $\operatorname{Rep}_\text f(G)$ generated by $V$ under tensor product and dual and direct sum, and the nice exercise implies that $\operatorname{Rep}_\text f(G)$ is the idempotent completion of this subcategory.

Furthermore, the Tannaka half of Tannaka–Krein duality tells us that $G$ is determined by $\operatorname{Rep}_\text f(G)$ in a suitable sense, although depending on how you take "suitable sense" means you may instead recover the complexification $G_{\mathbb{C}}$. From here on I will blithely ignore the difference between $G$ and its complexification. (Really I should say something here about averaging over a compact group and $\operatorname U(n)$ being the maximal compact subgroup of $\operatorname{GL}_n(\mathbb{C})$.)

Taken together, these two results tell us that $G$ or maybe its complexification is determined as a subgroup of $\operatorname{GL}(V)$ by its $G$-invariant tensors $\operatorname{Hom}(V^{\otimes m}, V^{\otimes n})^G$. What this means is that we ought to be able to define various $G$ of interest by saying "the $G$ preserving such-and-such tensors," and we do.

Moreover, if we decompose a given space of tensors $\operatorname{Hom}(V^{\otimes m}, V^{\otimes n})$ into its irreducible components under the action of $GL(V)$, then $G$ preserves some tensor iff it preserves the projection of the tensor to each irreducible component, so we can restrict our attention to collections of "irreducible tensors."

The tensors of rank $1$ are not so interesting; the stabilizer of a nonzero vector $v \in V$ is a general affine group, so we don't get anything new. Next are the tensors of rank $2$. The tensors in $\operatorname{Hom}(V, V)$ are again not so interesting; generically their stabilizers look like products of $\operatorname{GL}(V_i)$ where $V_i$ are the eigenspaces of a diagonalizable $T \in \operatorname{Hom}(V, V)$, so we again don't get anything new. So the next candidate is bilinear forms, and since $V^{\otimes 2} \cong \operatorname S^2(V) \oplus \bigwedge^2(V)$ is the irreducible decomposition here, we are naturally led to considering the stabilizers of symmetric resp. skew-symmetric forms, hence to the orthogonal and symplectic groups.

It's not just bilinear stuff out there though; to get the special linear groups we have to go all the way to a tensor in $V^{\otimes \dim V}$, namely any choice of a nonzero element of $\bigwedge^{\dim V}(V)$, and we can get, for example, $G_2$ using trilinear forms. But bilinear stuff is the simplest stuff after linear stuff.

According to a theorem of Serre, all semi-simple Lie groups are linear algebraic groups. See https://en.wikipedia.org/wiki/Complex_Lie_group for the precise statement and a reference.

That shows why we should look at "algebraic functions" when looking for complex semi-simple Lie groups. Natural place to start is subgroups of $GL(n, \mathbb{C})$ that preserve some linear forms. But then we get something isomorphic to $GL(n, \mathbb{C})$ as such subgroup has to preserve the kernel. Bilinear forms are the next best things and it turns out they provide plenty of examples. But we don't have to stop there! The complex Lie group $F_4$ can be defined as the subgroup of $GL(26, \mathbb{C})$ fixing a symmetric trilinear form. And the complex simple Lie group $G_2$ can be defined as the stabilizer of a generci $3$-form on $\mathbb{C}^7.$ There are similar descriptions for $E$-series. See e.g. this answer by Robert Bryant https://mathoverflow.net/a/99795/6818 (I think he wrote about this more explicitly somewhere else on MO, but I have trouble finding it. Anyway... this description of $E$-series goes back to Elie Cartan.)