Ambivalent not implies strongly ambivalent

Statement
It is possible to have an ambivalent group (i.e., a group in which all elements are fact about::real elements) that is not a strongly ambivalent group. In particular, there is at least one element that is not a fact about::strongly real element.

As a corollary, a real element of a group need not be a strongly real element.

Example of the quaternion group
The quaternion group of order eight is an ambivalent group. However, it is not strongly ambivalent, because the only elements of order $$2$$ are $$\pm 1$$, and these definitely do not generate the whole group.

Other examples
Other examples include particular example::special linear group:SL(2,5).