The algebra of $W$-invariant polynomial funktions sl2

Question

Let $L=\mathfrak{sl}(2,\mathbb{F})$, $H$ a borel subalgebra, $\Delta=\{\alpha\}$ a base of the corresponding root system and $W$ the Weyl group.
Let $\lambda=\frac{1}{2}\alpha$ be the fundamental dominant weight.
I read, that $\lambda^{2}$ generates the algebra of $W$-invariant polynomial functions $\mathfrak{P}(H)^{W}$, but I don´t get, why this should be true.
I though of the following:
I already know that algebra of polynomial functions $\mathfrak{P}(H)$ is generated by the $\lambda^{k}$ $k\in\mathbb{Z}^{+}$ and that $W$ is spaned by the reflexion $\sigma_{\alpha}$, so it is enought to check, what happenes, when $\sigma_{\alpha}$ acts on the $\lambda^{k}$. Since $\sigma_{\alpha}$ sends $\alpha$ to $-\alpha$ it sends $\lambda$ to $-\lambda$, so this is not an element of $\mathfrak{P}(H)^{W}$.
But I do not know how to continue for $k\geq2$.
Thank you for helping me.