Choose basis ( X a ) {\displaystyle (X_{a})} , a ∈ { 1 , 2 , ⋯ , dim ( G ) } {\displaystyle a\in \{1,2,\cdots ,\dim({\mathcal {G}})\}} for G {\displaystyle {\mathcal {G}}} and basis ( e a ) {\displaystyle (e_{a})} , a ∈ { 1 , 2 , ⋯ , dim ( R ) } {\displaystyle a\in \{1,2,\cdots ,\dim(R)\}} for R {\displaystyle R}
Notation: Let [ X a , X b ] = F a , b n X n {\displaystyle [X_{a},X_{b}]=F_{a,b}^{n}X_{n}} , F a b n ∈ Q {\displaystyle F_{ab}^{n}\in \mathbb {Q} }
< X a , X b >= t a b {\displaystyle <X_{a},X_{b}>=t_{ab}} ; t a b {\displaystyle t^{ab}} is defined s.t. t a b ⋅ t b c = δ a c {\displaystyle t_{ab}\cdot t^{bc}=\delta _{ac}}
, 
for 
and basis 
, 
for 
![{\displaystyle [X_{a},X_{b}]=F_{a,b}^{n}X_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48876a09c495ff70f7cf818fed5bd205ce716a37)
, 
; 
is defined s.t. 
