The following paragraph is from page 150 in the book Representation Theory by Fulton and Harris.
Observe that when we exponentiate the image of $sl_2 \mathbb{C}$ under the embedding $sl_2\mathbb{C} \rightarrow sl_{n+1}\mathbb{C}$ corresponding to the representation $\mathrm{Sym}^n V$, we arrive at the group $SL_2 \mathbb{C}$ when $n$ is odd and $PGL_2 \mathbb{C}$ when $n$ is even. Thus, the representation of the group $PGL_2 \mathbb{C}$ are exactly the even powers $\mathrm{Sym}^{2n}V$.
Here $V$ is the standard representation of $sl_2\mathbb{C}$ and $\mathrm{Sym}^n$ denotes the symmetric $n$th power.
I want to understand the embedding $sl_2\mathbb{C} \rightarrow sl_{n+1}\mathbb{C}$. This cannot be the obvious one because then we would get $SL_2\mathbb{C}$ for every $n$. There should be an embedding corresponding to $\mathrm{Sym}^n V$. How can I find this embedding?