  We derive a general result about commuting actions on certain objects in
braided rigid monoidal categories. This enables us to define an action of the
Brauer algebra on the tensor space $V^{\otimes k}$ which commutes with the
action of the orthosymplectic Lie superalgebra $\spo(V)$ and the
orthosymplectic Lie color algebra $\spo(V,\beta)$. We use the Brauer algebra
action to compute maximal vectors in $V^{\otimes k}$ and to decompose
$V^{\otimes k}$ into a direct sum of submodules $T^\lambda$. We compute the
characters of the modules $T^\lambda$, give a combinatorial description of
these characters in terms of tableaux, and model the decomposition of
$V^{\otimes k}$ into the submodules $T^\lambda$ with a Robinson-Schensted-Knuth
type insertion scheme.
