  We prove, for any infinite field k, that any virtually trivial split
spherical BN-pair in the group G(k) of k-rational points of a reductive k-group
G is already trivial. We then inspect the case when G is k-anisotropic and show
that in many situations G(k) admits no non-trivial split spherical BN-pairs.
This improves results and contributes to a conjecture of Caprace and Marquis,
which can be viewed as a converse to a well-known result of Borel and Tits.

Group Theory