Since $\Omega X$ is a $H$-space, if it has homology of finite type, the homology acquires the structure of a Hopf algebra. Bott has shown that for $X=G$ a Lie group, in fact $H_*(\Omega X)$ is free and concentrated in even degree.

Bott has also, using generating varieties, calculated this Hopf algebra structure for $G = SO(n)$, $Spin(n)$, $SU(n)$, $G_2$ in *The space of loops on a Lie group. Michigan Math. J., 5:35-61, 1958*. The same technique was used by Watanabe in *The homology of the loop space of the exceptional group F4. Osaka J. Math., 15:463-474, 1978* for $F_4$ and by Nakagawa in *The space of loops on the exceptional Lie group E6. Osaka J. Math., 40:429-448, 2003* for $E_6$. Using Bott periodicity, Kono and Kozima calculated this Hopf algebra structure on the homology for $Sp(n)$ in *The space of loops on a symplectic group. Japanese J. Math., 4:461-480, 1978*.

However, I haven't been able to locate similar calculations for the exceptional Lie groups $E_7$ and $E_8$. Does anyone know where these can be found? Seeing that it is already quite involved to do these calculations for $F_4$ and $E_6$, I'd rather not try to them myself. However, if some simple method exists, answers explaining it are also welcome.

Also note that Bott's result for $G_2$ in *The space of loops on a Lie group* seems to be incorrect. Watanabe writes in *The homology of the loop space of the exceptional group F4*:

There is a misprint in Bott's result on $H_\ast(\Omega G_2)$ [5;p. 60]. The coproduct formula for $w \in H_{10}(ΩG_2)$ is an error. It is corrected by exchanging 2 for 3.

This can also be found in Clarke's *On the K -theory of the loop space of a Lie group, Proc. Camb. Phil. Soc. (1974), 76, 1*.