Embedding the ring of algebraic integers into $\mathbb{R}^n$ (Serge Lang Algebra Exercise 7.4)

There is a difference between real embeddings and complex embeddings. Notice that if $\sigma$ is an embedding from $L$ to $\mathbb{C}$, then $\overline{\sigma}$ is also an embedding and $\sigma=\overline{\sigma}$ if and only if $\sigma$ is a real embedding.

If we denote $\sigma_{1},\dots,\sigma_{r_{1}}$ for those real embeddings $L\rightarrow \mathbb{R}$ and $\sigma_{r_{1}+1},\overline{\sigma_{r_{1}+1}},\dots,\sigma_{r_{1}+r_{2}},\overline{\sigma_{r_{1}+r_{2}}}$ for complex embeddings. Then the map should be $$\alpha\mapsto (\sigma_{1}(\alpha),\dots,\sigma_{r_{1}}(\alpha),\operatorname{Re}\sigma_{r_{1}+1}(\alpha),\operatorname{Im}\sigma_{r_{1}+1}(\alpha),\dots,\operatorname{Re}\sigma_{r_{1}+r_{2}}(\alpha),\operatorname{Im}\sigma_{r_{1}+r_{2}}(\alpha)),$$ or $$\alpha\mapsto (\sigma_{1}(\alpha),\dots,\sigma_{r_{1}}(\alpha),\sigma_{r_{1}+1}(\alpha),\dots,\sigma_{r_{1}+r_{2}}(\alpha))$$ if we view $\mathbb{C}$ as $\mathbb{R}^{2}$. In this way we can show that $\mathfrak{o}_{L}$ is a lattice of rank $r_{1}+2r_{2}=n$.