This is one of a series of the author’s papers on certain root-systems closely related to the root-systems associated with certain classes of Lie algebras. This particular paper is very closely related to a paper [Algebras Groups Geom. 6, No.2, 113-133 (1989; Zbl 0703.17005)] in which he studied the regular subsymmetrysets of a Jacobi set R. Here he defines and studies weak regular subsymmetrysets of R and shows that they are just the regular subsymmetrysets of R. (Weak regular symmetrysets are generated by weak regular sets \(\pi =\{a_ 1,...,a_ n\}\), which are defined by dropping one of the conditions required of regular sets, thus leaving the conditions (1) \(a_ i-a_ j\not\in R\) for all \(i\neq j\), and (2) there exists no infinite sequence \(c_ 1,c_ 2,...,c_ s,..\). of elements of R with \(c_ j-c_{j-1}\in \pi\) for all j.)