Klein Four-Group as Subgroup of S4

Theorem
Let $G$ be the following subset of the symmetric group on $4$ letters $S_4$, expressed in two-row notation:

Then $G$ is an example of the Klein $4$-group.

Proof
By inspection, the Cayley table is constructed:


 * $\begin{array}{c|cccc}

& e & a & b & c \\ \hline e & e & a & b & c \\ a & a & e & c & b \\ b & b & c & e & a \\ c & c & b & a & e \\ \end{array}$

Again by inspection this can be seen to be the same as the Cayley table for the Klein $4$-group.