I came across this question when attempting exercise 10.10 in Nielson and Chaung's Quantum Computation and Quantum Information. The exercise was "Explicitly verify the quantum error-correction conditions for the Shor code, for the error set containing $I$ and the error operators $X_{j}, Y_{j}, Z_{j}$ for $j = 1, 2, 3, 4, 5, 6, 7, 8, 9$".
My attempt so far was to encode $$|0\rangle \rightarrow \frac{1}{2\sqrt{2}}(|000\rangle + |111\rangle) \otimes (|000\rangle + |111\rangle) \otimes (|000\rangle + |111\rangle)$$
and
$$|1\rangle \rightarrow \frac{1}{2\sqrt{2}}(|000\rangle - |111\rangle) \otimes (|000\rangle - |111\rangle) \otimes (|000\rangle - |111\rangle)$$
I think the projector $P$ is $P = |+++\rangle\langle+++| + |---\rangle\langle---| $
I am not sure about this last part and I'm not really sure where to go from here. I think that the goal is to show that if $\{E_{i}\}$ are the error operators then $$PE_{i}^{*T}E_{j}P = \alpha _{ij}P$$ for $\alpha _{ij}$ is a complex hermitean matrix.