Proving equivalence of two Turing machines
The main idea behind your proof is correct.
Note that (assuming the tape is infinitely sized on the right only) in the operator algebra of tape movements on the right-infinite tape we have identy $xR^kyL^kz = xyz$.
Using that we can prove:
$$R = R^4L^3 = (R^2)^2L^3$$
$$L = R^2L^3 $$
And since the right hand side consists only of unit moves in your model, you can do the effective moves on the left hand side. And you already know how, you replace every transition $T(x, S_i) = (y, S_o, m)$ where $m \in \{L,R\}$ with series
\begin{align}
&amp;&amp;T(x, S_i) &amp;= (y, s_1(m), m'_1(m))\\
&amp;\forall z\in \Sigma :\quad &amp;T(z, s_1(m)) &amp;= (z, s_2(m), m'_2(m))\\
&amp;&amp;&amp;\cdots\\
&amp;\forall z\in \Sigma :\quad &amp;T(z, s_{k-1}(m)) &amp;= (z, S_o, m'_k(m))
\end{align}
where $s_i$ are new temporary states and $m'_i$ are the appropriate series of moves from $\{R^2, L^3\}$.