TITLE: Bilinear forms in compact/semisimple Lie group theory
QUESTION [8 upvotes]: If you look up the list of compact or semisimple Lie groups, you will see that three out of four infinite families (B, C and D) are defined in terms of a bilinear form on a vector space, either symmetric or skew-symmetric.
Are there any underlying reasons for this prominence of bilinear/quadratic forms in Lie group theory? Why do they, and not any other geometric objects, play such a fundamental role?

REPLY [8 votes]: 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.)