Unique involution in SL2 over an odd field ($A^2=I,\det A=1⟹A=\pm I$)

Over a field of odd characteristic, an involutory matrix has minimal polynomial dividing $(X-1)(X+1)$, so it is diagonalizable with eigenvalues in $\{1,-1\}$. In dimension two, determinant one requires either both eigenvalues one or both minus one. Hence the only element of order two in $S{L}_2$ is $-I$; the identity has order one.