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We present the formulae for twist quantization of $g_2$, corresponding to thesolution of classical YB equation with support in the 8-dimensional Borelsubalgebra of $g_2$. The considered chain of twists consists of the fourfactors describing the four steps of quantization: Jordanian twist, the twotwist factors extending Jordanian twist and the deformed Jordanian or in secondvariant additional Abelian twist. The first two steps describe as well the$sl(3)$ quantization. The coproducts are calculated for each step in expliciteform, and for that purpose we present new formulas for the calculation ofsimilarity transformations on tensor product. We introduce new basic generatorsin universal enveloping algebra $U(g_2)$ which provide nonlinearities inalgebraic sector maximally simplifying the deformed coproducts.Comment: LaTeX, aps, jmp class, 24 pages. Minor changes, the version in press  in JM

Basic twist quantization of the exceptional Lie algebra g2