On the image of a word map with constants of a simple algebraic group II

This paper is a continuation of the investigations of images of word maps with constants $\widetilde{w}_\Sigma: G^n \rightarrow G$ on a simple algebraic group $G$ started in the work of F. Gnutov and N. Gordeev, On the image of a word map with constants of a simple algebraic group, Zap. Nauchn. Semin. POMI, 478 (2019), 78–99. In this paper we prove that for adjoint simple algebraic groups of the type $B_l, C_l, F_4, G_2$ over a field of characteristic $\ne 2, 3$ the map $\pi\circ \widetilde{w}$, where $\widetilde{w}_\Sigma$ is word map without small constants and $\pi: G\rightarrow T/W$ is a map of factorization, is a constant map if and only if $w_\Sigma=vgv^{-1}$, where $g \in G$ and $v$ is a word with constants.
Also, we give estimates for dimensions of images of some types of word maps with constants on simple algebraic groups.