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The paper continues the study of images of word maps with constants r w Σ : G n → G of a simple algebraic group G, started by F. Gnutov and N. Gordeev (2019). It is proved that for adjoint simple algebraic groups of type B l , C l , F 4 , G 2 over a field of characteristic = 2, 3, the map π • r w, where r w Σ is word map without small constants and π : G → T /W is the factorization map, is a constant map if and only if w Σ = vgv −1 , where g ∈ G and v is a word with constants. Bibliography: 9 titles.