Uniqueness of addition in Lie algebras of Chevalley type over rings with $1/2$ and $1/3$

In this paper, it is proved that Lie algebras of Chevalley type ($A_n$, $B_n$, $C_n$, $D_n$, $E_6$, $E_7$, $E_8$, $F_4$, and $G_2$) over associative commutative rings with $1/2$ (with $1/2$ and $1/3$ in the case of $G_2$) have unique addition. As a corollary of this theorem, we note the uniqueness of addition in semisimple Lie algebras of Chevalley type over fields of characteristic ${\ne}\, 2$ (${\ne}\, 2,3$ in the case of $G_2$).