Identifying Lorentzian conformal group after radial quantization

Consider radial quantization in Euclidean signature, in $d$ dimensions. We obtain a Hilbert space with a (non-Hermitian) representation of the Euclidean conformal algebra $so(1,d+1)$, with these commutation relations (taking $eta_{munu}$ Euclidean). One view of "Wick rotation" is that we keep the same formal Hilbert space, but we use complex linear combinations of the $so(1,d+1)$ generators to obtain a Hermitian representation of the Lorentzian conformal algebra $so(2,d)$. How do we obtain the Hermitian $so(2,d)$ generators from complex linear combinations of the $so(1,d+1)$ generators? (And how could we have seen abstractly that this must be possible? It should follow because both Lie algebras are "real forms" of $so(d+2)$)?

In comments here Peter Kravchuk says this is possible, and it probably boils down to a simple computation after choosing the right linear combinations. He says

If you go to a basis in which [the generators] are hermitian (by
taking combinations like $P+K$ and $i(P−K)$) then you will be able to
identify the commutation relations with $SO(2,d),$ not $SO(1,d+1).$

However I will provide more background, because I also want to corroborate the general story.

Background: Equal-time quantization.
Let’s first discuss equal-time quantization in Euclidean signature, i.e. quantizing using the $t=0$ slice. We obtain obtain generators with the same commutation relations linked above, again taking $eta_{munu}$ Euclidean. (I’ll ignore $K_mu$ and $D_mu$, just focusing on Euclidean isometries for now.) We find that the generator of Euclidean time-translations is anti-Hermitian, $P_0^dagger = -P_0$, thus generating positive operators under physicists’s exponentiation $exp(i cdot)$. Meanwhile generators of spatial translations are Hermitian (thus generating unitary operators), and $M_{0nu}$ are anti-Hermitian. So our representation of the Euclidean isometry group is not unitary. However, if we define new generators $P_0 mapsto iP_0$, $M_{0nu} mapsto iM_{0nu}$ (and leave the remaining generators unchanged), we obtain a set of Hermitian generators that satisfy the commutation relations but now with Lorentzian $eta_{munu}$. (This is can be checked by tracking factors of $i$.) That is, we obtain a unitary representation of Poincare.

Radial quantization:
Now consider radial quantization in Euclidean space. I believe we again obtain generators with the same commutation relations, again taking $eta_{munu}$ Euclidean. However, the adjoint (Hermitian conjugation) is defined differently in radial quantization than in usual (equal-time) quantization. In radial quantization, conjugation corresponds to inversion, and we obtain $D=-D^dagger,$ $M_{munu}=M_{munu}^dagger,$ and $P_mu = K_mu^dagger$. (Is this right?) See e.g. David-Simmons Duffin, Eq. 112. (Note his generators should differ by a factor of $i$ from the above, due to his convention of exponentiating without a factor of $i$.) If I understand correctly, the expectation is that we can now take linear combinations of generators, such as $iD$, $P_mu+K_mu$ and $i(P_mu-K_mu)$, that will be Hermitian and that will satisfy the $so(2,d)$ commutation relations. But which combinations do we choose to form the standard $so(2,d)$ generators?

Example approach:
First note that in general, given the usual generators $J_{ab}$ (for $a,b=1,…,p+q$) of a representation of $so(p,q)$, i.e. satisfying $J_{ab}=-J_{ba}$ and
$$ [J_{mu nu },J_{rho sigma }]=i(eta _{nu rho }J_{mu sigma }+eta _{mu sigma }J_{nu rho }-eta _{mu rho }J_{nu sigma }-eta _{nu sigma }J_{mu rho })$$ where $eta$ has signature $(p,q)$, we can form a representation of $so(p+1,q-1)$ using new generators $tilde{J}_{ab} = i^s J_{ab}$, where $s=0$ unless $a=p+1$ or $b=p+1$. That is, if we want to flip the sign in the signature of the $p+1$ axis, we use generators that differ by a factor of $i$ whenever they involve the $p+1$ axis.

Now given the Euclidean generators discussed under "radial quantization" above, we want to form Hermitian generators $D_L$, $(P_mu)_L$, $(K_mu)_L$, and so on (using subscript $L$ for "Lorentzian"). By writing the usual $D,P,K,M$ generators of $so(1,d+1)$ generators in terms of the $J$ generators of $so(1,d+1)$, we should be able to use the above method to find the appropriate $so(2,d)$ generators, and then re-write these in terms of $D,P,K, M$ generators for $so(2,d+1)$. However, we need to check the generators we obtain this way are Hermitian. I think we end up with generators like
begin{align}
D_L & overset{?}{=} frac{1}{2}(P_0 + K_0)
(P_i)_L & overset{?}{=} frac{1}{2}(P_i + K_i) – M_{i0},
(K_i)_L & overset{?}{=} frac{1}{2}(P_i + K_i) + M_{i0},
(M_{mu,0})_L & overset{?}{=} frac{1}{2} i(K_mu-P_mu)
end{align}
and so on? If this works out, all the Lorentzian generators should be Hermitian and have the right commutation relations for the Lorentzian conformal algebra.

Finishing this computation and checking the hermiticity of the resulting generators is probably straightforward… but is there a reference for this basic aspect of radial quantization?