  The main goal of this paper is to introduce and explore an appropriate notion
of weakly Rickart JB$^*$-triples. We introduce weakly order Rickart
JB$^*$-triples, and we show that a C$^*$-algebra $A$ is a weakly (order)
Rickart JB$^*$-triple precisely when it is a weakly Rickart C$^*$-algebra. We
also prove that the Peirce-2 subspace associated with a tripotent in a weakly
order Rickart JB$^*$-triple is a Rickart JB$^*$-algebra in the sense of Ayupov
and Arzikulov. By extending a classical property of Rickart C$^*$-algebras, we
prove that every weakly order Rickart JB$^*$-triple is generated by its
tripotents.
