On the sum of the index of a parabolic subalgebra and of its nilpotent radical

In this short note, we investigate the following question of Panyushev stated in 2003: ``Is the sum of the index of a parabolic subalgebra of a semisimple Lie algebra and the index of its nilpotent radical always greater than or equal to the rank of ?' Using the formula for the index of parabolic subalgebras conjectured by Tauvel and the author and proved by Fauquant-Millet and Joseph in 2005 and Joseph in 2006, we give a positive answer to this question. Moreover, we also obtain a necessary and sufficient condition for this sum to be equal to the rank of . This provides new examples of direct sum decomposition of a semisimple Lie algebra verifying the ``index additivity condition' as stated by Raïs.