I have been trying to understand Nogueira answer to this question
He states a very interesting way to constrain a complexified Lie algebra to a real one, through what he calls "Reality conditions". I've never seen this anywhere and would like a more in depth understanding of this, especially how you find reality conditions for any specific algebras which complexify to the same algebra.
There’s a good discussion of this in the books of J.F. Cornwell.
The key point of the answer you linked to is that there are many real algebras that go to the same complex form (all the $so(p,q)$ for instance) but that going back to real forms is usually a delicate process.
Basically, by going to the complexified version, you allow the diagonal operator to be... well.. diagonal with complex eigenvalues (with the usual math convention). This is what we do for instance in angular momentum theory when we choose eigenstates of $L_z$, i.e. when we go to a basis where $L_z$ is diagonal. This is some technicality because eigenvalue equations can be solved over the complex that cannot be solved over the reals.
Coming back to angular momentum, the “real” matrix realization would not be one where the exponentiation of $L_z$ is diagonal but rather something like \begin{align} R_z(\theta)\to \left( \begin{array}{ccc} \cos\theta&-\sin\theta & 0 \\ \sin\theta&\cos\theta &0 \\ 0&0&1\end{array}\right) \end{align} clearly not diagonal.
For compact groups, going to the complexified algebra is not a problem because decomplexifying does not change any of the results: representations irreducible under the complex extension remain irreducible when decomplexified. It’s a matter of going back to the original generator by simply undoing the complex combinations.
The situation changes considerably with non-compact groups, such as Lorentz. There is no guarantee that the representation irreducible under the real version are irreducible under the complexified version. For instance, the adjoint representation of Lorentz is irreducible over the reals, but reduced to $(1,0) \oplus (0,1)$ over the complex field.
In practice the generators of the real forms are the rotations and boosts (as real matrices), and you cannot take complex combos like $L_j+i K_j$ that allow you to break up the adjoint of Lorentz in 2 commuting sets of operators having angular momentum like commutation relations. The generators of $so(4)_\mathbb{R}$ can be arranged as a sum of $su(2)$’s under complexification, but returning to the real form will not give you something that is like the representations of the Lorentz, $so(3,1)$. (Clearly $so(4)_\mathbb{R}$ is not the same as $so(3,1)$.)
The unitary representations of the non-compact groups are also infinite-dimensional, so that the usual tensor construction for irreps of the complexified Lorentz cannot possibly become unitary under decomplexification.
There’s a cute “tutorial” paper
Campoamor-Stursberg R, et al. $su (2)$-expansion of the Lorentz algebra $so (3, 1)$. Canadian Journal of Physics. 2013;91(8):589-98,
wherein the authors construct real (infinite-dimensional) representations of Lorentz and where the subtleties between real and complexified form is presented in the context of Lorentz with physics-level rigour.
