Creation and annihilation operator gymnastics

Consider a two level system described by fermionic creation and annihilation operators. \[\begin{aligned} &\{c_{i\sigma}^\dagger,c_{j\sigma'}^{\phantom{\dagger}}\}=\delta_{ij}\delta_{\sigma\sigma'} \\ &\{c_{i\sigma}^\dagger,c_{j\sigma'}^\dagger\}=\{c_{i\sigma}^{\phantom{\dagger}},c_{j\sigma'}^{\phantom{\dagger}}\}=0 \end{aligned}\] for \(i,j=1,2\) and \(\sigma,\sigma'=\uparrow\downarrow\)
 
a.) Construct the total spin operators \(\hat{S}^z,\hat{S}^+,\) and \(\hat{S}^-\). Verify, using the second quantized form that \[[\hat{S}^+,\hat{S}^-]=2\hat{S}^z, \ [\hat{S}^z,\hat{S}^\pm]=\pm\hat{S}^\pm\] b.) The pseudospin operators are \[\begin{aligned} &\hat{J}^+=c_{1\uparrow}^\dagger c_{1\downarrow}^\dagger-c_{2\uparrow}^\dagger c_{2\downarrow}^\dagger \\ &\hat{J}^-=c_{1\downarrow}^{\phantom{\dagger}}c_{1\uparrow}^{\phantom{\dagger}}-c_{2\downarrow}^{\phantom{\dagger}}c_{2\uparrow}^{\phantom{\dagger}} \\ &\hat{J}_z=\frac{1}{2}\left(c_{1\uparrow}^\dagger c_{1\uparrow}^{\phantom{\dagger}}+c_{1\downarrow}^\dagger c_{1\downarrow}^{\phantom{\dagger}}+c_{2\uparrow}^\dagger c_{2\uparrow}^{\phantom{\dagger}}+c_{2\downarrow}^\dagger c_{2\downarrow}^{\phantom{\dagger}}\right)-1 \end{aligned}\] Show the pseduospin operators satisfy the \(SU(2)\) algebra: \[[\hat{J}^+,\hat{J}^-]=2\hat{J}^z, \ [\hat{J}^z,\hat{J}^\pm]=\pm\hat{J}^\pm\] Also, show that the spin and psuedospin operators commute. That is, \[[\hat{J}^\pm,\hat{S}^\pm]=[\hat{J}^\pm,\hat{S}^z]=[\hat{J}^z,\hat{S}^\pm]=[\hat{J}^z,\hat{S}^z]=0\] c.) Find \(J,m_J,s,m_s\) for the following states: \[c_{1\uparrow}^\dagger\left|0\right\rangle, \ c_{1\uparrow}^\dagger c_{2\uparrow}^\dagger\left|0\right\rangle, \ \frac{1}{\sqrt{2}}(c_{1\uparrow}^\dagger c_{2\downarrow}^\dagger\pm c_{1\downarrow}^\dagger c_{2\uparrow}^\dagger)\left|0\right\rangle, \ \frac{1}{\sqrt{2}}(c_{1\uparrow}^\dagger c_{1\downarrow}^\dagger-c_{2\uparrow}^\dagger c_{2\downarrow}^\dagger)\left|0\right\rangle\]