$$
M=
\begin{pmatrix}
-\sin 0 & -\sin \frac{2\pi}{3} & -\sin -\frac{2\pi}{3}\\
\cos 0 & \cos \frac{2\pi}{3} & \cos -\frac{2\pi}{3}\\
\frac{1}{3} & \frac{1}{3} & \frac{1}{3}
\end{pmatrix}=
\begin{pmatrix}
0 & -\frac{\sqrt{3}}{2} & \frac{\sqrt{3}}{2}\\
1 & -\frac{1}{2} & -\frac{1}{2}\\
\frac{1}{3} & \frac{1}{3} & \frac{1}{3}
\end{pmatrix}
$$
$$
M^{-1}=
\begin{pmatrix}
0 & \frac{2}{3} & 1\\
\frac{1}{\sqrt{3}} & -\frac{1}{3} & 1\\
-\frac{1}{\sqrt{3}} & -\frac{1}{3} & 1
\end{pmatrix}
$$
$$
R(\theta)=
\begin{pmatrix}
\cos\theta & -\sin\theta & 0\\
\sin\theta & \cos\theta & 0\\
0 & 0 & 1
\end{pmatrix}
$$
$$
R(\theta)^{-1}=R(-\theta)
$$
$$
\begin{pmatrix}
v_{x0}\\
v_{y0}\\
L\omega
\end{pmatrix}=
R(\theta)
\begin{pmatrix}
v_x\\
v_y\\
L\omega
\end{pmatrix}=
R(\theta)M
\begin{pmatrix}
v_1\\
v_2\\
v_3
\end{pmatrix}
$$
$$
\begin{pmatrix}
v_1\\
v_2\\
v_3
\end{pmatrix}
= M^{-1}
\begin{pmatrix}
v_x\\
v_y\\
L\omega
\end{pmatrix}
= M^{-1}R(-\theta)
\begin{pmatrix}
v_{x0}\\
v_{y0}\\
L\omega
\end{pmatrix}
$$
