  For the split group $G_{2}$ defined over $\mathbb{Z},$ we show that the de
Rham cohomology ring of $B(G_{2})_{\mathbb{F}_{2}}$ is isomorphic to the
singular cohomology ring with $\mathbb{F}_{2}$-coefficients of
$B(G_{2})_{\mathbb{C}}.$ For the spin groups $\textrm{Spin}(n)$ defined over
$\mathbb{Z},$ we show that the de Rham cohomology ring of
$B\textrm{Spin}(n)_{\mathbb{F}_{2}}$ is isomorphic to the singular cohomology
ring with $\mathbb{F}_{2}$-coefficients of $B\textrm{Spin}(n)_{\mathbb{C}}$ for
$n \leq 10.$ For $n=11,$ we make a full computation of the de Rham cohomology
ring of $B\textrm{Spin}(11)_{\mathbb{F}_{2}},$ which is not isomorphic to the
singular cohomology ring with $\mathbb{F}_{2}$-coefficients of
$B\textrm{Spin}(11)_{\mathbb{C}}.$ We also show that the Hodge spectral
sequence for $BG_{\mathbb{F}_{2}}$ degenerates for all of the groups $G$
mentioned above.
