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It is shown that for a higher weak isospin symmetry, $SU(P)_L$ with $P\geq 3$, the baryon minus lepton charge $B-L$ neither commutes nor closes algebraically with $SU(P)_L$ similar to the electric charge $Q$, which all lead to a $SU(3)_C\otimes SU(P)_L\otimes U(1)_X\otimes U(1)_N$ gauge completion, where $X$ and $N$ determine $Q$ and $B-L$, respectively. As a direct result, the neutrinos obtain appropriate masses via a canonical seesaw. While the version with $P=3$ supplies the schemes of single-component dark matter well established in the literature, we prove in this work that the models with $P\geq 4$ provide the novel scenarios of multicomponent dark matter, which contain simultaneously at least $P-2$ stable candidates, respectively. In this setup, the multicomponet dark matter is nontrivially unified with normal matter by gauge multiplets, and their stability is ensured by a residual gauge symmetry which is a remnant of the gauge symmetry after spontaneous symmetry breaking. The three versions with $P=4$ according to the new lepton electric charges are detailedly investigated. The mass spectrum of the scalar sector is diagonalized when the scale of the $U(1)_N$ breaking is much higher than that of the usual 3-4-1 symmetry breaking. All the interactions of gauge bosons with fermions and scalars are obtained. We figure out viable parameter regimes given that the multicomponent dark matter satisfies the Planck and (in)direct detection experiments.