To a certain extent, Deligne's exceptional series $A_1 \subset A_2 \subset G_2 \subset D_4 \subset F_4 \subset E_6 \subset E_7 \subset E_8$ plays a role analogous to the classical series $A_n \subset A_{n+1} \subset \dots \subset SL(\infty)$, $C_n \subset C_{n+1} \subset \dots \subset Sp(\infty)$, and $D_n \subset B_n \subset D_{n+1} \subset \dots \subset Spin(\infty)$.

One place the classical series occur is in constructions of K-theory. In particular, it is approximately true that if you apply the Quillen plus construction to $BSU(\infty)$, you get complex K-theory, and if you look instead at $\mathbb{F}$-points of $GL(\infty)$, you get algebraic K-theory of the field $\mathbb{F}$. I imagine that similar constructions starting with $Spin$ and $Sp$ give versions of real K-theory, but I don't know.

**Question:** What is the result of Quillen's plus construction applied to $BE_8$? Does it produce an "exceptional K-theory"?