I am following Nielsen and Chuang, section 2.2.5:
A projective measurement is described by an observable, $M$, a Hermitian operator on the state space of the system being observed. The observable has a spectral decomposition, $$M = \sum_m mP_m$$ where $P_m$ is the projector onto the eigenspace of $M$ with eigenvalue $m$. The possible outcomes of the measurement correspond to the eigenvalues, $m$, of the observable.
My question is how can we view a the projective measurement of two qubits in the standard (Z) basis using this formalism. In this measurement the projectors are $$\left|00\right\rangle \left\langle 00\right|, \left|01\right\rangle \left\langle 01\right|, \left|10\right\rangle \left\langle 10\right|, \left|11\right\rangle \left\langle 11\right|$$
But how can I find them if I only know of the observable, which in this case is $Z\otimes Z$ if I understand correctly?
This observable has only two eigenvalues: $+1$ and $-1$. The eigenvalue $+1$ corresponds both to $\left|00\right\rangle$ and $\left|11\right\rangle$, so it seems to me I need a "stronger" observable, which can differentiate those two states, or I'm missing a something elementary in the definition of projective measurements.