Let $G$ and $H$ be two groups. a reduced word is an element of the form $w=g_1h_1g_2h_2...$ after removing an instance of the identity element (of either $G$ or $H$) and replacing a pair of the form $g_1g_2$ by its product in $G$, or a pair $h_1h_2$ by its product in $H$. Denote $G*H$ the set of reduced words $w$ on which we define a group law by concatenation of words and we call it the free product of $G$ and $H$.
when $G=H$ a word in $G*G$ is an element $w=g_1^1g_2^2g_3^1g_4^2...$ where $g_i^j$ is the element $g_i$ from the $j$th copy, $j=1,2$. Why we are not allowed to replace $g_1^1g_1^2g_2^1g_2^2...$ by the element $g$ who is the product $g_1^1g_2^2g_3^1g_4^2...$ since product $g_k^1g_l^2$ is well defined being in the same group $G$ ?