  We prove two results about $\text{SLF}(\bar U_q)$, the algebra of symmetric
linear forms on the restricted quantum group $\bar U_q = \bar
U_q(\mathfrak{sl}(2))$. First, we express any trace on finite dimensional
projective $\bar U_q$-modules as a linear combination in the basis of
$\text{SLF}(\bar U_q)$ constructed by Gainutdinov - Tipunin and also by Arike.
In particular, this allows us to determine the symmetric linear form
corresponding to the modified trace on projective $\bar U_q$-modules. Second,
we give the explicit multiplication rules between symmetric linear forms in
this basis.
