  For a general Multidimensional L\'{e}vy process (satisfying some moment
conditions), we introduce the Multidimensional power jump processes and the
related Multidimensional Teugels martingales. Furthermore, we orthogonalize the
Multidimensional Teugels martingales by applying Gram-Schmidt process. We give
a chaotic representation for every square integral random variable in terms of
these orthogonalized Multidimensional Teugels martingales. The predictable
representation with respect to the same set of Multidimensional orthogonalized
martingales of square integrable random variables and of square integrable
martingales is an easy consequence of the chaotic representation.
