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Given a quaternion division algebra D, a noncentral element e ∈ D × is called pure if its square belongs to the center. A theorem of Rowen and Segev (2004) asserts that for any quaternion division algebra D of positive characteristic > 2 and any pure element e ∈ D × the quotient D × /X(e) of D × by the normal subgroup X(e) generated by e, is abelian-by-nilpotent-by-abelian. In this note we construct a quaternion division algebra D of characteristic zero containing a pure element e ∈ D such that D × /X(e) contains a nonabelian free group. This demonstrates that the situation in characteristic zero is very different.

Nonabelian free subgroups in homomorphic images of valued quaternion division algebras