Automorphism theorem

Question

Help me please to find reference for the proof of the following theorem:

Theorem. Let $\theta$ be a Leibniz cocycle on the Leibniz algebra L with values in $V,$ and assume $\theta^{\bot}\cap C(L)=(0).$ Then the automorphism group $Aut(L_{\theta})$ of the extension algebra $L_{\theta}$ consists of all linear operators of the matrix form $$ \left[ \begin{array}{lr} \alpha_0 & 0 \\ \\ \phi & \psi \\ \end{array} \right], $$

where $\alpha_0 \in Aut(L), \ \psi = \alpha|{C(L_{\theta})} \in GL(k),$ and $\phi \in Hom(L, V),$ such that $\theta (\alpha_0x, \alpha_0y) = \phi[x,y] + \psi\theta(x,y),$ all $x, y \in L.$

(L, [.,.]) is a Leibniz algebra over F

Answer 1

I doubt the reference for exactly such statement exists. The closest reference I am aware of is: D. Liu, L. Lin, On the toroidal Leibniz algebras, Acta Math. Sinica 24 (2008), N2, 227-240 DOI:10.1007/s10114-007-1003-z . There, in Propostion 5.1, they establish a relationship between automorphisms of a perfect Leibniz algebra and automorphisms of its universal central extension, and Corollary 5.3 appears to be a particular case of your statement. The arguments are quite straitforward, repeating almost verbatim the corresponding arguments in the case of Lie algebras due to A. Pianzola (see references in that paper), and, if true, it seems not to be difficult to extend them to your statement.