  We show that in a weak commutative inverse property loop, such as a Bruck
loop, if $\alpha$ is a right [left] pseudoautomorphism with companion $c$, then
$c$ [$c^2$] must lie in the left nucleus. In particular, for any such loop with
trivial left nucleus, every right pseudoautomorphism is an automorphism and if
the squaring map is a permutation, then every left pseudoautomorphism is an
automorphism as well. We also show that every pseudoautomorphism of a
commutative inverse property loop is an automorphism, generalizing a well-known
result of Bruck.

Group Theory