  Greenberger-Horne-Zeilinger (GHZ) theorem asserts that there is a set of
mutually commuting nonlocal observables with a common eigenstate on which those
observables assume values that refute the attempt to assign values only
required to have them by the local realism of Einstein, Podolsky, and Rosen
(EPR). It is known that for a three-qubit system there is only one form of the
GHZ-Mermin-like argument with equivalence up to a local unitary transformation,
which is exactly Mermin's version of the GHZ theorem. In this paper, however,
for a four-qubit system which was originally studied by GHZ, we show that there
are nine distinct forms of the GHZ-Mermin-like argument. The proof is obtained
by using some geometric invariants to characterize the sets of mutually
commuting nonlocal spin observables on the four-qubit system. It is proved that
there are at most nine elements (except for a different sign) in a set of
mutually commuting nonlocal spin observables in the four-qubit system, and each
GHZ-Mermin-like argument involves a set of at least five mutually commuting
four-qubit nonlocal spin observables with a GHZ state as a common eigenstate in
GHZ's theorem. Therefore, we present a complete construction of the GHZ theorem
for the four-qubit system.
