Three qubit identity

Problem:

I’m having trouble seeing how this paper claims the following identity to be true for a three qubit system (labelled by $A$, $B$, and $C$) under a pure state $|psi rangle$ with real coefficients

$$langle X_{A} X_{B}rangle ^2 + langle Z_{A} Z_{B}rangle ^2 + langle Z_{A} X_{B}rangle ^2 + langle X_{A} Z_{B}rangle ^2 = 1 + langle Y_{A} Y_{B}rangle ^2 – langle Y_{A} Y_{C}rangle ^2 – langle Y_{B} Y_{C}rangle ^2$$

where $X$, $Y$, and $Z$ are the three Pauli matrices and the subscript is the qubit label. The expectation values have the usual meaning $langle X_A X_B rangle equiv text{tr} rho X_A otimes X_B otimes 1$ where $rho equiv text{tr}_C |psi rangle langle psi |$.

My Guess:

I’m thinking that this identity arises out of something like $X^2 + Y^2 + Z^2=3$ but that approach leads me nowhere since the square is outside the expectation brackets. Another problem is that while on the LHS the $C$ space is being traced over, on the RHS it’s either $A$ or $B$ that’s being traced over, and the square on top of everything just makes it worse. I suspect this is a well known but tediously derived identity, in which case I’ll be happy to be pointed to some source material which grinds this out.

Thanks in advance. Cheers.