In Fulton and Harris 'Representation Theory: A first course', page 150, we have the following paragraph
"Observe that when we exponentiate the image of $\mathfrak{sl}_2(\mathbb{C})$ under the embedding $\mathfrak{sl}_2(\mathbb{C}) \rightarrow \mathfrak{sl}_{n+1}(\mathbb{C})$ corresponding to the representation $\text{Sym}^nV$, we arrive at the group $\text{SL}_2(\mathbb{C})$ when $n$ is odd and $\text{PGL}_2(\mathbb{C})$ when $n$ is even."
Now I am not sure how to explicitely find this embedding, and I cannot see how it is evident that we get different groups depending on the parity of $n$. Can someone explain to me how one can see that?