The authors consider the normality question of formally normal operators. They answer the question under some condition which is related with the commutant of a formally normal operator. Developing the idea of {\it B. Sz.-Nagy} [Extensions of linear transformations in Hilbert spaces which extend beyond this space, Appendix to {\it F. Riesz, B. Sz.-Nagy}, Functional analysis (1960; for the French original see

Zbl 0046.331)] in the case of unbounded operators they adopt for their purpose a dilation theory of positive definite forms over a *-semi-group. This leads in a natural way to spectral (integral) representations of these forms as well as to the normality question of formally normal operators. Applying those results for unbounded subnormal operators they find conditions reminding the well known {\it P. R. Halmos} characterization [Summa Brasil. Math. 2, 125-134 (1950;

Zbl 0041.232)].