I have a problem with the proof of this quaternions property, included below.
$(q^∗)^∗ = [s, −(−v)] = [s, v] = q$， $$q^∗q = (q^∗)(q^∗)^∗= ∥q^∗∥^2= s^2 + x^2 + y^2 + z^2= ∥q∥^2= qq^∗$$ so we got $q^*q = qq^*$.
Why in the last equation we lost one of the conjugate marks?
Conjugate operation is involutive, so $(q^*)^* = q$.
Thus, since $qq^*$ is real, $qq^* = (qq^*)^*=q^*(q^*)^*=q^*q$.