Cellularity for weighted KLRW algebras of types B$B$, A(2)$A boolean AND{(2)}$, D(2)$D boolean AND{(2)}$

This paper constructs homogeneous affine sandwich cellular bases of weighted KLRW algebras in types BZ > 0$B_{\mathbb {Z}_{\geqslant 0}}$, A2 center dot e(2)$A<^>{(2)}_{2\cdot e}$, De+1(2)$D<^>{(2)}_{e+1}$. Our construction immediately gives homogeneous sandwich cellular bases for the finite-dimensional quotients of these algebras. Since weighted KLRW algebras generalize KLR algebras, we also obtain bases and cellularity results for the (infinite and finite-dimensional) KLR algebras.