An extension of the original implementation of JADE, named eJADE((1)) hereafter, was proposed in 2001 to perform independent component analysis for any combination of statistical orders greater than or equal to three. More precisely, eJADE((1)) relies on the joint diagonalization of a set of several cumulant matrices corresponding to different matrix slices of one or several higher order cumulant tensors. An efficient way, without lose of statistical information, of reducing the number of third and fourth order cumulant matrices to be jointly diagonalized is proposed in this paper. The resulting approach, named eJADE(3,4)((2)), can be interpreted as an improvement of the eJADE(3,4)((1)) method. A performance comparison with classical methods is conducted in the context of MRS and EEG signals showing the good behavior of our technique.