It is a version of a question I asked on Math.StackExchange (no answers yet). Suppose $a_1,a_2,\ldots,a_n,b$ is a primitive system in a non-abelian free group $F$ (a part of a basis of $F$). Let $c \in F'$ be an element of the commutator subgroup of $F.$ Is there any simple condition on $c$ for the set $$ a_1,a_2,\ldots,a_n, bc $$ NOT to be a primitive system in $F$? An interesting case is the case where $bc$ is a primitive element of $F.$

Granted, for any given $c \in F',$ we can verify by means of the classical algorithm whether the set generates a free factor of $F,$ or we can reduce the problem to invertibility of a suitable matrix over the group ring $\mathbf Z[F].$