3 fixed matrix

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

2 typo in title

AutomorphizmAutomorphism theorem

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 automorphizm 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

1

Automorphizm theorem

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 automorphizm 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