  The stabilizer ZX-calculus is a rigorous graphical language for reasoning
about quantum mechanics.The language is sound and complete: a stabilizer
ZX-diagram can be transformed into another one if and only if these two
diagrams represent the same quantum evolution or quantum state. We show that
the stabilizer ZX-calculus can be simplified, removing unnecessary equations
while keeping only the essential axioms which potentially capture fundamental
structures of quantum mechanics. We thus give a significantly smaller set of
axioms and prove that meta-rules like 'colour symmetry' and 'upside-down
symmetry', which were considered as axioms in previous versions of the
language, can in fact be derived. In particular, we show that the additional
symbol and one of the rules which had been recently introduced to keep track of
scalars (diagrams with no inputs or outputs) are not necessary.
