  Let $n\in\mathds{N}$ and $B_n(r,q)$ be the generic Birman-Murakami-Wenzl
algebra with respect to indeterminants $r$ and $q$. It is known that $B_n(r,q)$
has two distinct linear representations generated by two central elements of
$B_n(r,q)$ called the symmetrizer and antisymmetrizer of $B_n(r,q)$. These
generate for $n\geq 3$ the only one dimensional two sided ideals of $B_n(r,q)$
and generalize the corresponding notion for Hecke algebras of type $A$. The
main result in this paper explicitly determines the coefficients of these
elements with respect to the graphical basis of $B_n(r,q)$.
