Lorentz algebra structure in Hilbert space vs Vector space representation "must have the same commutation relations"

I’m following the Srednicki book, Ch. 2

The definitions are:

Small Lorentz transformation on Minkowski space, so $S^{mu nu}$ forms the vector representation of the Lorentz group
$$
Lambda^rho_{;;tau} = delta^rho_tau + frac{i}{2 hbar} delta omega_{mu nu} (S^{mu nu})^rho_{;;tau}, quad (S^{mu nu})^rho_{;;tau} = frac{hbar}{i} left( g^{mu nu} delta^nu_{;;tau} – g^{nu rho} delta^mu_tau right)
$$

Small Lorentz transformation on Hilbert space, so $M^{mu nu}$ are generators of the Lorentz group on Hilbert space and $Lambda to U (Lambda)$ is group homomorphism.
$$
U(1 + delta omega) = I + frac{i}{2 hbar} delta omega_{mu nu} M^{mu nu}
$$

We also proved the following earlier (commutation relations between $M$‘s)
$$
[M^{mu nu}, M^{rho sigma}] = i hbar left( g^{mu rho} M^{nu sigma} – (mu leftrightarrow nu) right) – (rho leftrightarrow sigma)
$$

These type of relations can be proven from
$$
U (Lambda)^{-1} M^{mu nu} U (Lambda) = Lambda^mu_{;;rho} , Lambda^nu_{;;sigma} M^{rho sigma}
$$

There is also a representation on functions (I haven’t found this stated explicitly in Srednicki, but it seems to follow from (2.26))
$$
L (1 + delta omega) varphi (x) overset{?}{=} varphi (x) – frac{i}{2 hbar} delta omega_{mu nu} mathcal{L}^{mu nu} varphi (x), quad mathcal{L}^{mu nu} = frac{hbar}{i} left( x^mu partial^nu – x^mu partial^nu right)
$$

My goal is to solve the problem 2.9 part b) using the hint: Show that the matrices $S^{mu nu}$ must have the same commutation relations as the operators $M^{mu nu}$. Hint: see the previous problem. (the hint is important)

Now we proved a lot of interesting stuff in the "previous problem"
$$
[varphi (x), M^{mu nu}] = mathcal{L}^{mu nu} varphi (x)
$$
$$
[[varphi (x), M^{mu nu}], M^{rho sigma}] = mathcal{L}^{mu nu} mathcal{L}^{rho sigma} varphi (x)
$$
$$
[varphi (x), [M^{mu nu}], M^{rho sigma}]] = left( mathcal{L}^{mu nu} mathcal{L}^{rho sigma} – mathcal{L}^{rho sigma} mathcal{L}^{mu nu} right) varphi (x)
$$

And from 2.9 a)
$$
[partial^rho varphi (x), M^{mu nu}] = mathcal{L}^{mu nu} partial^rho varphi (x) + (S^{mu nu})^rho_{;;tau} partial^tau varphi (x)
$$

And perhaps most interestingly, the last one hints that
$$
partial^rho mathcal{L}^{mu nu} = (S^{mu nu})^rho_{;;tau} partial^tau
$$
i.e. differentiating expressions with $mathcal{L}$ also reveals $S$.

From how the question is formulated with that hint, it seems like I can somehow derive in a clever way that the commutator of $S$‘s is of the same form as of the $M$‘s, but I can’t think of any other way than just plugging in the explicit form given earlier. I tried picking various forms for $varphi (x)$ (like $(Lambda^{-1} x)^mu$ and use the last commutator) but I didn’t get anything useful.

Is there really some clever way to show the commutation relations must be the same, or is it only done by brute force and crunching indices? What is the hint for? Why does it say to see the previous problem?

On the one hand I think that it would be rather unreasonable to define $M^{mu nu}$, $S^{mu nu}$ or even $mathcal{L}^{mu nu}$ in a way that they don’t have the same commutator, because they essentially do the same transformation, just on different objects (elements of Hilbert space, Minkowski or functions on spacetime). On the other hand, the commutation relation (even while keeping the metric tensor preserved) boils down to details like factors of $hbar$ and $i$ (we can for example multiply the generator by 2 and the metric tensor is still preserved), so it is not unreasonable that the book demands a separate verification.