Prove that if matrices (T_{a}) with matrix elements (left(T_{a} ight)_{b c}) proportional to the structure constants (f_{a b c}) are defined as in Eq. (3.8),

Prove that if matrices \(T_{a}\) with matrix elements \(\left(T_{a}\right)_{b c}\) proportional to the structure constants \(f_{a b c}\) are defined as in Eq. (3.8), these matrices satisfy the Lie algebra (3.9). Thus, show that the structure constants generate a representation of the algebra with dimension equal to the number of generators.

Data from Eq. (3.8)

Data from Eq. (3.9)

(Ta)bc (Xb Ta |Xc) = -ifabc, =