Some reductive anisotropic groups that admit no non-trivial split spherical BN-pairs

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.