In my textbook (in french), the formula of mixed model is written as:
$y = X \beta + Z u + \epsilon$
Here, the author says that $Z$ should be a sub-vector of $X$, without explaining further why.
His example is a mixed model for repeated measures, where fixed effect are
intercept + X + Y + time + X:time and random effects are
individual_ID + time. As he presents the example, he considers
individual_ID as an intercept.
However, I could not find this information anywhere else. Indeed, in numerous courses and tutorials (this one for instance), there can be random effects (group) which are not included in fixed effects.
Is this rule commonly accepted?
If it is, what is the problem of indluding a random effect without its fixed effect? Are above-mentioned tutorials wrong?
If it is not, why would the author of my textbook write something like this?