Seeking for a converse to a well-known theorem by Borel-Tits, we address the
question whether the group of rational points G(k) of an anisotropic reductive
k-group may admit a split spherical BN-pair. We show that if k is a perfect
field or a local field, then such a BN-pair must be virtually trivial. We also
consider arbitrary compact groups and show that the only abstract BN-pairs they
can admit are spherical, and even virtually trivial provided they are split.