  In this article we consider conditions under which projection operators in
multiplicity free semi-simple tensor categories satisfy Temperley-Lieb like
relations. This is then used as a stepping stone to prove sufficient conditions
for obtaining a representation of the Birman-Murakami-Wenzl algebra from a
braided multiplicity free semi-simple tensor category. The results are found by
utalising the data of the categories. There is considerable overlap with the
results found in arXiv:1607.08908, where proofs are shown by manipulating
diagrams.
