  We elaborate an approach to quantum fluctuations of angular momentum based on
the diagonalization of the covariance matrix in two versions: real symmetric
and complex Hermitian. At difference with previous approaches this is SU(2)
invariant and avoids any difficulty caused by nontrivial commutators.
Meaningful uncertainty relations are derived which are nontrivial even for
vanishing mean angular momentum. We apply this approach to some relevant
states.
