A note on the special unitary group of a division algebra

If $D$ is a division algebra with its center a number field $K$ and with an involution of the second kind, it is unknown if the group $SU(1,D)/[U(1,d)$, $U(1,D)]$ is trivial. We show that, by contrast, if $K$is a function field in one variable over a number field, and if $D$ is an algebra with center $K$ and with an involution of the second kind, the group $SU(1,D)/[U(1,d),U(1,D)]$ can be infinite in general. We give an infinite class of examples.