An eight-dimensional nilpotent Leibniz algebra without outer derivations

Jacobson's classical theorem asserts that every nilpotent Lie algebra admits a non-inner derivation. This result, however, does not extend to Leibniz algebras: there exists an eight-dimensional complex nilpotent Leibniz algebra whose every derivation is inner. In this paper, we present a systematic investigation of this algebra, with particular emphasis on the absence of outer derivations and its structural properties. We describe its lower central series, annihilators, center, and Leibniz kernel, and prove that its derivation algebra is a six-dimensional nilpotent Lie algebra isomorphic to the quotient of the algebra by its Leibniz kernel. We further show that its automorphism group is a connected six-dimensional unipotent group obtained by exponentiating inner derivations; in particular, it contains no nontrivial elements of finite order. In addition, we provide a complete classification of the quasiderivation, generalized derivation, and $σ$-derivation of the algebra. Finally, we show that, among the four linearly independent infinitesimal deformations, exactly one is integrable, giving rise to a one-parameter family of pairwise non-isomorphic algebras which, with a single exception, also admit only inner derivations.