An　improved　adjoint　operator　method　is　employed　to　compute　third　order　normal　forms　of　an　eight-dimensional　nonlinear　dynamical　system　and　the　associated　nonlinear　transformation　for　the　first　time.　Two　different　cases　which　respectively　are　four　pairs　of　pure　imaginary　eigenvalues　and　one　non-semisimple　double　zero　and　three　pairs　of　pure　imaginary　eigenvalues　for　the　linear　operator　are　considered　in　the　eight-dimensional　nonlinear　dynamical　system.　First,　an　improved　adjoint　operator　method　is　described　in　detail　by　analyzing　the　eight-dimensional　nonlinear　dynamical　system.　The　formulae　of　computing　third　order　normal　forms　of　the　eight-dimensional　nonlinear　system　are　derived　and　the　Maple　symbolic　program　is　developed　in　two　different　cases　of　the　linear　operator.　Then,　third　order　normal　forms　and　their　coefficients　for　the　eight-dimensional　nonlinear　dynamical　system　in　two　different　cases　of　the　linear　operator　are　obtained　by　executing　the　Maple　program.　The　relationship　between　the　coefficients　of　normal　forms　and　ones　of　the　original　nonlinear　systems　is　given.　Finally,　this　approach　is　also　applied　to　obtain　normal　form　of　the　averaged　equation　for　a　parametrically　excited　viscoelastic　moving　belt.　The　results　obtained　here　indicate　that　this　improved　adjoint　operator　method　is　a　convenient　and　efficient　approach　to　obtain　normal　forms　of　higher　dimensional　nonlinear　dynamical　systems.　It　is　also　shown　that　we　may　respectively　obtain　normal　forms,　their　coefficients　and　the　associated　near　identity　nonlinear　transformations　for　the　eight-dimensional　nonlinear　system　in　two　different　cases　by　using　a　same　main　Maple　symbolic　program.