It would be nice to extend our discussion to the last remaining exceptional Lie group, namely $E_8$. However, unlike the other exceptional Lie groups, $E_8$ does not have any “small” representations; the minimal representation of $E_8$ is the adjoint representation, with 248 elements. The minimal representation of $G_2$ is $\OO$ (7-dimensional), $F_4$ acts on the trace-free part of the Albert algebra (26-dimensional), $E_6$ acts the Albert algebra (27-dimensional), and $E_7$ acts on $\{\Pc=(\XXX,\YYY,p,q)\}$, that is on two copies of the Albert algebra, together with two real numbers (56-dimensional). But $E_8$ acts on nothing smaller than itself.

The geometry of $E_8$ is therefore beyond the scope of this book.