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

In this paper we consider some properties of a word map with constants $\tilde{w}: G^n \rightarrow G$ of a simple algebraic groups $G$ and some properties of maps $\pi \circ \tilde{w}$, where $\pi:G\rightarrow T/W$ is the factor morphism for a fixed maximal torus $T$ of the group $G$ and the Weil group $W$ of $G$. In particular, we prove here that for an adjoint group $G$ of the types $A_r, D_r, E_r$ the map $\pi\circ \tilde{w}$ is a constant map only for words of the type $v g v^{-1}$ where $g \in G$ and $v$ is a word with constants. The corollary of this result is the following generalization of the result of T. Bandman and Yu. G. Zarhin ( Eur. J. Math. 2 (2016), 614–643): the image of a word map with constant $\tilde{w}: \mathrm{PGL}_2^n \rightarrow \mathrm{PGL}_2$ contains a representation of every semisimple conjugacy class $\ne 1$ or $w = vgv^{-1}$ for some $g, v$.