I have been studying a course on Lie algebras in particle physics and I could never understand how complexifying helps us understand the *original* Lie algebra. 

For example, consider $\mathfrak{su}(2)$: I complexify this to give me $\mathfrak{su}(2)\_\mathbb{C}$ which allows me to form a Cartan-Weyl basis of ladder operators and a Cartan subalgebra, so I can generate highest weight representations. This is fine, but these are representations of $\mathfrak{su}(2)\_\mathbb{C}$ not $\mathfrak{su}(2)$ because we cannot form a Cartan-Weyl basis unless we complexify. So how does this help us construct representations of $\mathfrak{su}(2)$?

All across particle physics we talk about particles living in representations of particular Lie algebras, but in fact should they actually be living in the *complexifications* of these? i.e. we talk about spinors of the Lorentz group, but the way to come across these is by complexifying the Lie algebra $\mathfrak{so}(3,1)$ so that it decomposes as

$$\mathfrak{so}(3,1)\_\mathbb{C}=\mathfrak{su}(2)\_\mathbb{C} \oplus \mathfrak{su}(2)\_\mathbb{C}.$$

At which point I can label representations by $(A,B)$ where $A,B$ label the highest weights of the two subalgebras, and I would say that left-handed spinors live in $(1/2,0)$ and right handed in $(0,1/2)$. But again, these are *complexified* Lie algebras. How does this tell me that spinors exist with respect to the real Lorentz group, the one which the universe uses.

In summary, my questions:

1. How does complexifying a Lie algebra $\mathfrak{g}$ to $\mathfrak{g}\_\mathbb{C}$ help me discover representations of $\mathfrak{g}$ when the highest weight method only works with complexified Lie algebras?
2. How do I know that the things which I discover after complexifying, like spinors and particle multiplets, are valid with respect to the orginal Lie algebra? i.e. We live in a universe where Lorentz transformations are real NOT complex, so how can we discover spinors without complexifying our Lorentz algebra?