Im a physics student an I have been studying Lie Groups and Lie algebras for some time from a mathematical point of view mostly following Hall's book. Thing is that the Highest Weight Theorem is ennunciated for complex semisimple Lie algebras and the irr. reps. are labeled with the weights that are of course positive integers. But in physics we work with $su(2)$ that is not a complex Lie algebra, and we label its irr. reps. by a Half Integer. In Hall it just says that
In the physics literature, the representations of $su(2)\cong so(3)$ are labeled by the parameter $l=m/2$. In terms of this notation, a representation of $so(3)$ comes from a representation of $SO(3)$ if and only if $l$ is an integer. The representations with $l$ an integer are called “integer spin”; the others are called “half-integer spin.”
So it just lets it as a notational thing. Thing is that this cannot be, Half integer in something key in physics, it just arises naturally. I thought it may be because the isomorphism $sl(2,\mathbb{C})\cong su(2)\oplus su(2)$ Like, i dont know, half integer + integer =integer but i dont even know how to show that, if true. Anything is largely apreciated.
To my mind, the explanation is at the level of Lie groups, not Lie algebras, since the Lie algebras preserve slightly less information than the Lie groups. Here, while $su(2)\to so(3)$ is an isomorphism, the corresponding Lie group map $SU(2)\to SO(3)$ is $2$-to-$1$. So only about half the irreducibles of $SU(2)$ descend to $SO(3)$. The ones that do descend are easily distinguished (as it turns out) by some parity issues (equivalently, integer/half-integer).
The other incidental issue in your question, about real versus complex Lie algebras, is not directly related to this... but perhaps it is helpful to note that a complex repn of a real Lie algebra has a canonical extension to a repn of the complexification of that Lie algebra on the same complex vector space. (And, it's not really the case that the complexification of $su(2)$ is two copies of it... more accurate is that it's $s\ell(2)$).
EDIT: yes, by $s\ell(2)$ I mean the complex Lie algebra. And, the way complexification of Lie algebras (or lots of other things over-the-reals) work does product two copies of the original but only as real vector spaces... not as Lie algebras, for example. As a simpler example, the complexification of the ring (or field) $\mathbb R$, is $\mathbb C$, as a ring/field. As a real vector space, sure, it's $\mathbb R\oplus \mathbb R$, but not as a ring/field.