Embeddings and $C^*$-envelopes of exact operator systems

We prove a necessary and sufficient condition for embeddability of an operator system into $\mathcal{O}_2$. Using Kirchberg's theorems on a tensor product of $\mathcal{O}_2$ and $\mathcal{O}_{\infty}$, we establish results on their operator system counterparts $\mathcal{S}_2$ and $\mathcal{S}_{\infty}$. Applications of the results proved, including some examples describing $C^*$-envelopes of operator systems, are also discussed.