We can then quotient L2 by this relation to get a vector space V
I think you’re confusing two different parts here. There’s a quotient of a vector space to get a vector space, which is done to embed $\mathcal{L}$ in a vector space. There’s also something sort of like a projectivization, which does not produce a vector space. In the method I prefer, there isn’t an explicit quotient, but instead just functions on the vector space that satisfy certain properties. (I could see being convinced to prefer the other version if it did improve the presentation.)
I think you’re confusing two different parts here. There’s a quotient of a vector space to get a vector space, which is done to embed $\mathcal{L}$ in a vector space. There’s also something sort of like a projectivization, which does not produce a vector space. In the method I prefer, there isn’t an explicit quotient, but instead just functions on the vector space that satisfy certain properties. (I could see being convinced to prefer the other version if it did improve the presentation.)