Adjoint representation of Liealgebra

\[we\,\,want\,\,to\,\,prove\,\,that \\ ad\left( ab \right) -ad\left( ba \right) =ad\left( a \right) ad\left( b \right) -ad\left( b \right) ad\left( a \right) \\ ad\left( ab \right) _{ij}=ad\left( a \right) _{ik}ad\left( b \right) _{kj} \\ \left[ ab,a_j \right] =\left[ a,a_k \right] ad\left( b \right) _{kj} \\ \left[ ab,a_j \right] -\left[ ba,a_j \right] =\left[ a,\left[ b,a_j \right] \right] -\left[ b,\left[ a,a_j \right] \right] \\ aba_j-a_jab-\left( baa_j-a_jba \right) =\left[ a,\left( ba_j-a_jb \right) \right] -\left[ b,aa_j-a_ja \right] \\ =aba_j\cancel{-aa_jb}\cancel{-ba_ja}+a_jba-baa_j\cancel{+ba_ja}\cancel{+aa_jb}-a_jab \\ so\,\,we\,\,have\,\,that\,\,ad\left( ab \right) -ad\left( ba \right) =ad\left( a \right) ad\left( b \right) -ad\left( b \right) ad\left( a \right) \\ but\,\,mind\,\,that\,\,we\,\,dont\,\,have\,\,ad\left( ab \right) =ad\left( a \right) ad\left( b \right) \]