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Every nontrivial quotient group of a group generated by two involutions is generated by two involutions, allowing repeated generators. The two generator images have squares equal to the identity. If one image becomes trivial, the other generates $C_2$; if both become trivial, the quotient is trivial and is excluded. Thus the two-involution characterization of a dihedral group is preserved by nontrivial quotients.

Let $a,b$ be the generating involutions. If $a\ne b$, set $x=ab$ and $y=b$. Then $x\ne e$, so its order of a group element is at least two or infinite, andTherefore $x,y$ generate the group and satisfy the required dihedral inversion relation. If $a=b$, the group is $C_2$ and $x=y=a$ gives its permitted degenerate description. This proves the two-involution characterization of a dihedral group without excluding either $C_2$ or the infinite dihedral group.

The two-involution characterization of a dihedral group uses the extended convention that includes $C_2$ and the infinite dihedral group; the two generators need not be distinct. Put $a=xy$ and $b=y$. The inversion relation givesIf $a\ne e$, both $a,b$ are involutions and generate the original group. If $a=e$, then $x=y$ and the group is $C_2$; choose the same nonidentity element twice. Thus two elements of order two generate the group, including the degenerate case allowed here. The distinction between square equal to the identity and order of a group element exactly two is essential.