  In the study of substitution minimal subshifts, some complicated trivialities
have hindered simple and general approaches. Recently, Maloney and Rust
introduced the term "tame," simplifying the study. We introduce another term
"$l$-primitive" for the substitutions and show that the combination of these
two conditions can characterize the minimality of substitution subshifts. We
shall show that all substitution minimal subshifts can be generated by
substitutions that satisfy both conditions; conversely, all substitutions that
satisfy the two conditions always generate minimal subshifts. As an
application, we show that the result by Damanik and Lenz that an admissible
substitution subshift is minimal if and only if it is linearly repetitive is
valid for all substitution subshifts. The above set of conditions can be
checked by finite calculations (algorithms).
