Background
I recently asked this question: Can this strange implicit matrix equation be solved?. Followed by Is this "fractal equation" for fractals constructed by finite subdivision rules complete?. However in the second question I realised I was imprecise and decided to think more carefully and ask a new question in the same spirit.
Describing a fractal
Let us describe the mapping $T$
$$T:\hat A \to \begin{pmatrix} \hat A \\ R_{-\pi/3} \hat A \\ R_{\pi/3}\hat A \\ \hat A \\ \end{pmatrix} =\hat A_1 $$
$$T * T : \hat A \to \begin{pmatrix} \hat A_1 \\ R_{-\pi/3} \hat A_1 \\ R_{\pi/3}\hat A_1 \\ \hat A_1 \\ \end{pmatrix} = \hat A_2 $$
Where $$R_\phi A_1 = \begin{pmatrix} R_\phi \hat A \\ R_\phi R_{-\pi/3} \hat A \\ R_\phi R_{\pi/3}\hat A \\ R_\phi \hat A \\ \end{pmatrix} $$
Where $R_{\theta}$ is a rotation matrix that obeys: $R_{\theta} \cdot R_{\phi} = R_{\theta+ \phi} $. Hence, upon infinitely applying $T$ and setting $\hat A$ to represent a vector, we have something that describes the below image:
Hence, we have a "fractal equation" for the above recursive picture. Also this gives one a very natural way to multiply fractals. For example:
We can define an operation:
$$ \hat A \otimes \begin{pmatrix} \hat 1 \\ R_{-\pi/3} \\ R_{\pi/3} \\ \hat 1 \\ \end{pmatrix} = \begin{pmatrix} \hat A \\ R_{-\pi/3} \hat A \\ R_{\pi/3}\hat A \\ \hat A \\ \end{pmatrix} $$
It's the same as outer product with the exception being the components of the matrix are matrices themselves.
$$ L: \hat B \to \hat B \otimes \begin{pmatrix} \hat 1 & \hat 1 \\ \hat 1 & \hat 0 \end{pmatrix} $$
which can describe this when $B$ is set to be a square:
Questions
Is this fractal equation complete (can it describe any Finite subdivision rules fractals)? How would one go about proving this (if true?)?