Let $R$ be a ring and $G$ be a group and $RG$ be the group ring. Denote by $[R,R]$, the additive subgroup generated by all lie products $[x,y]=xy-yx , \forall\ x,y\in R$.
Then how is this that $[\sum_{g\in G}\beta(g)g,\sum_{h\in G}\gamma(h)h]=\sum_{g,h}\beta(g)\gamma(h)[g,h]=\sum_{g,h}\beta(g)\gamma(h)(gh-hg)$.
If suppose I take two elements $x=a_1g_1+a_2g_2 , y=b_1g_1+b_2g_2$ in $RG$, then $[x,y]= (a_1g_1+a_2g_2)(b_1g_1+b_2g_2)-(b_1g_1+b_2g_2)(a_1g_1+a_2g_2)=(a_1b_1-b_1a_1)g_1^{2}+(a_2b_2-b_2a_2)g_2^{2}+(a_1b_2-b_1a_2)g_1g_2+(a_2b_1-b_2a_1)g_2g_1$.
But I am unable to put them in above required form. Any help is appreciated.
Thanks