As the title suggests, I need to show that the Witt algebra $W(1)$ with basis $\{e_i \, | \, -1 \leq i \leq p-2\}$ where $e_k=t^{k+1}\frac{\mathrm{d}}{\mathrm{d}t}$ with Lie bracket defined by
$$[e_i,e_j]=(j-i)e_{i+j} \, \, \, \, \, \, \, \, \, \, i+j \leq p-2$$ $$[e_i,e_j]=0 \, \, \, \, \, \, \, \, \, \, i+j \geq p-1,$$
is isomorphic to $\mathfrak{sl}(2,\mathbb{k})$ with basis $\{e,h,f\}$ with Lie brackets defined as
$$[h,e]=2e, \,\,[e,f]=h, \,\,[h,f]=-2f.$$
Now take $\mathrm{char}(\mathbb{k})=p=3$. The basis for $W(1)$ is then $\{e_{-1},e_0,e_1\}$ and the Lie brackets are
$$[e_0,e_{-1}]=-e_{-1}, \,\, [e_{-1},e_1]=2e_0, \,\,[e_0,e_1]=e_1.$$
I began by defining a linear map $\varphi: \mathfrak{sl}(2, \mathbb{k}) \rightarrow W(1)$ by
$$\varphi(e)=e_{-1}, \,\, \varphi(h)=e_0, \,\, \varphi(f)=e_1.$$
I then looked at $\varphi([e,h])=\varphi(-2e)=-2e_{-1}$. But this is not equal to $[\varphi(e),\varphi(h)]=[e_{-1},e_0]=e_{-1}$ and so clearly this map is not isomorphic. I'm unsure what map I need to use to show the two Lie algebras are isomorphic, could anyone give me any advice?