  A construction of bases for cell modules of the Birman--Murakami--Wenzl (or
B--M--W) algebra $B_n(q,r)$ by lifting bases for cell modules of $B_{n-1}(q,r)$
is given. By iterating this procedure, we produce cellular bases for B--M--W
algebras on which a large abelian subalgebra, generated by elements which
generalise the Jucys--Murphy elements from the representation theory of the
Iwahori--Hecke algebra of the symmetric group, acts triangularly. The
triangular action of this abelian subalgebra is used to provide explicit
criteria, in terms of the defining parameters $q$ and $r$, for B--M--W algebras
to be semisimple. The aforementioned constructions provide generalisations, to
the algebras under consideration here, of certain results from the Specht
module theory of the Iwahori--Hecke algebra of the symmetric group.
