# Is this unipotent group, over characteristic 2, connected?

Let $$E_{ij}(x)\in Mat_{7\times7}(\overline{\mathbb{F}}_2)$$ be the matrix with zeros everywhere, except for the value $$x$$ at $$ij$$. Set $$a(x)=1+E_{12}(x)+E_{34}(x)+E_{56}(x)$$, $$b(y)=1+E_{23}(y)+E_{45}(y)+E_{67}(y)$$, for two indeterminants $$x,y$$. Set $$G_8=$$. Is $$G_8$$ connected?

For easier visualization, here are the matrices:

$$a(x)=\begin{pmatrix} 1&x& & & & & \\ &1&0& & & & \\ & &1&x& & & \\ & & &1&0& & \\ & & & &1&x& \\ & & & & &1&0\\ & & & & & &1 \end{pmatrix} ,\ b(y)=\begin{pmatrix} 1&0& & & & & \\ &1&y& & & & \\ & &1&0& & & \\ & & &1&y& & \\ & & & &1&0& \\ & & & & &1&y\\ & & & & & &1 \end{pmatrix}$$