Summary: We prove that if \(\zeta\) is a semi-simple Lie bundle over a topological space X and \(\zeta =\epsilon_ 1\oplus...\oplus \epsilon_ m=\eta_ 1\oplus...\oplus \eta_ k\) (direct Whitney sum) where \(\epsilon_ i\) and \(\eta_ j\) are ideals which are simple Lie bundles, then \(m=k\) and the \(\eta\) ’\({}_ js\) coincide with the \(\epsilon\) ’\({}_ is\) (except for the order).