Dihedral Group D4/Cayley Table/Coset Decomposition of (e, a^2)

Cayley Table for Dihedral Group $D_4$
The Cayley table for the dihedral group $D_4$, with respect to the coset decomposition of the normal subgroup $\gen {a^2}$, is:

can be presented as:
 * $\begin{array}{l|cc|cc|cc|cc}

&    e &   a^2 &     a &   a^3 &     b & b a^2 &   b a & b a^3 \\ \hline e    &     e &   a^2 &     a &   a^3 &     b & b a^2 &   b a & b a^3 \\ a^2  &   a^2 &     e &   a^3 &     a & b a^2 &     b & b a^3 &   b a \\ \hline a    &     a &   a^3 &   a^2 &     e & b a^3 &   b a &     b & b a^2 \\ a^3  &   a^3 &     a &     e &   a^2 &   b a & b a^3 & b a^2 &     b \\ \hline b    &     b & b a^2 &   b a & b a^3 &     e &   a^2 &     a &   a^3 \\ b a^2 & b a^2 &    b & b a^3 &   b a &   a^2 &     e &   a^3 &     a \\ \hline b a  &   b a & b a^3 & b a^2 &     b &   a^3 &     a &     e &   a^2 \\ b a^3 & b a^3 &  b a &     b & b a^2 &     a &   a^3 &   a^2 &     e \end{array}$

which is seen to be an example of the Klein $4$-group.