We compute Grothendieck's torsion index of the compact Lie group for any . We explain the applications of the torsion index to topology and to the study of splitting fields for quadratic forms.
"The torsion index of the spin groups." Duke Math. J. 129 (2) 249 - 290, 15 August 2005. https://doi.org/10.1215/S0012-7094-05-12923-4