Spectral conditions for the nilpotency of Lie algebras

In the paper is proved that each of the following conditions is sufficient for the nilpotency of a solvable Lie algebra. (They are formulated by means of the spectrum of a Banach space representation of a Lie algebra, introduced by C. Ott in Pacific J.Math. 173(1996), 173-180.)

1) The validity of the projection property of the spectrum on Lie subalgebras (imposed to only one representation).

2) The existence of representations with spectra of arbitrary small diameters (with respect to a fixed norm on the given Lie algebra).