 Solution

Write $s=g_1$ and $r=g_2$. Direct matrix multiplication givesEvery product reduces to $r^k{s}^{\ell }$, with $0\le k<4$ and $0\le \ell <2$. These eight matrices are distinct and are precisely the listed elements:This is the dihedral group of group order eight. Its element $r^2$ commutes with both generators. Conjugation by $s$ interchanges $r$ and $r^{-1}$, while conjugation by $r$ sends $s$ to $r^{2}s$. It also interchanges $rs$ and $r^{-1}s$. The resulting conjugacy classes areThese are the stated $C_0,\dots ,C_4$ in order. To see that the last two reflection classes do not merge, conjugation changes the exponent in $r^{k}s$ by an even integer or changes its sign, preserving its parity. The five sets exhaust all eight elements.

Both generators are real orthogonal matrices with determinant $+1$. Hence every element obeys $g^{†}g=I_3$ and $\det g=1$. The normal-form calculation establishes closure and inverses, soIn particular the embedding in the special unitary group uses the given real matrices viewed as complex ones.