In this note we wish to implement some very technical consequences due to professor(s) P.J. Basser and Sinisa Pajevic \cite{5b} to discuss the spectral decomposition of 4th-order Douglas tensor in $^*$P-Finsler manifold. Actually, throughout the article, we have just reviewed the results of \cite{5b} for a particular 4th-order covariance  Douglas tensor, most often enunciated in Finslerian geometry. Moreover, spectral decomposition techniques have been studied for isotropic Douglas tensor.