  In this paper we give an alternative basis, $\mathcal{B}_{\rm ST}$, for the
Kauffman bracket skein module of the solid torus, ${\rm KBSM}\left({\rm
ST}\right)$. The basis $\mathcal{B}_{\rm ST}$ is obtained with the use of the
Tempereley--Lieb algebra of type B and it is appropriate for computing the
Kauffman bracket skein module of the lens spaces $L(p, q)$ via braids.
