  Let $A$ be a Banach algebra with a right identity $u$ such that $uA$ is
commutative and semisimple. In this paper, we investigate symmetric
bi-derivations of $A$ and detremine their range. We also study symmetric
bi-derivations of $A$ with their $k$-centralizing trace. Finally, we prove
every symmetric Jordan bi-derivation of $A$ is a symmetric bi-derivation.
