We compute Grothendieck's torsion index of a compact Lie group for all simply connected groups and all groups of adjoint type. In particular, the torsion index of the group is .
"The torsion index of and other groups." Duke Math. J. 129 (2) 219 - 248, 15 August 2005. https://doi.org/10.1215/S0012-7094-05-12922-2