Question on spin-orbit interaction in He I

I’m currently learning spin-orbit interaction and running into a problem to calculate the energy splitting in a neutral helium atom. For instance, I want to calculate how much the energy splittings are when the atom is at ${^{3}P_{J}}$ state. And here’s my understanding:

• Because of the symmetrization postulate, the ${^{3}P_{J}}$ wavefunction
  begin{equation}
  |psiranglein SpanBigg{frac{1}{sqrt2}(|100rangleotimes|21m_lrangle – |21m_lrangleotimes|100rangle)otimeschi_{sym} Bigg}
  end{equation}
  which is in a 9-dimensional space.

• By applying
  begin{equation}
  hat{S_{z1}}otimes I + Iotimeshat{S_{z2}}
  end{equation}
  I found that $m_sin{-1, 0, 1}$; thus, $S = 1$. Similarly, I found the total azimuthal quantum number $L = 1$ of the two electrons. Therefore, the total angular momentum $J = 0, 1, 2$, and they correspond to a 5-dimensional, a 3-dimensional and an 1-dimensional subspace.

• Then, as I understand it, I can use the 2-electron spin-orbit interaction Hamiltonian
  begin{equation}
  hat{H}_{SO} = frac{xi(r_1)}{2}(hat{J^2_1}otimes I – hat{L^2_1}otimes I – hat{S^2_1}otimes I) +
  frac{xi(r_2)}{2}(Iotimeshat{J^2_2} – Iotimeshat{L^2_2} – Iotimeshat{S^2_2})
  end{equation}
  to compute the energy splitting if I know the eigenstates that diagonalize it.

At this point, I want to first find the energy of
begin{equation}
frac{1}{sqrt2}(|100rangleotimes|211rangle – |211rangleotimes|100rangle)otimes|uparrowuparrowrangle
end{equation}
which is an eigenstate of $hat{H}_{SO}$ w. $J = 2$. However, I’m wondering how I could find the qunatum number $j_1$ of $hat{J^2_1}otimes I$ only? Is it the total angular momentum of a single electron state
begin{equation}
frac{1}{sqrt{2}}(|100rangle-|211rangle)otimes|uparrowrangle
end{equation}
If not, what is the correct procedure? Btw, is there a relation between $j_1, j_2$ and $J$? Thanks a lot!