Normal subgroup

Let $\mathcal{S}_4$ denote the symmetric $4$ - permutation group. Prove that

$$\mathcal{H}=\big\{{{\rm{id}},\,({1\,2})\,({3\,4}),\,({1\,3})\,({2\,4}),\,({1\,4})\,({2\,3})}\big\}$$

is a normal subgroup of $\mathcal{S}_4$.

Solution

Spoiler

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The interested reader may find two (not necessarily different) solutions here.