Abstract: A. G. Earnest and J. Hsia proved an arithmetical version of Springer's theorem for lattices in certain quadratic spaces. This result depends on the so called norm principle. In this work, we prove a similar result for skew-hermitian lattices. We also introduce a cohomology theory for the group of units of a lattice, to give a cohomological interpretation of the relation between norm principle and Springer type results, which we generalize to arbitrary semi-simple groups.