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Here is an example of a birational involution of $\mathbb P^n$ that is at least not obviously a combination of Cremona transformations:

Suppose $H\subset \mathbb P^n$ P^{n+1}$ is a hypersurface of degree $d$ in $\mathbb P^{n+1}$ such that it contains two points $P_1,P_2\in H$ such that the multiplicity of $H$ at $P_i$ is exactly $d-1$ for $i=1,2$. Such hypersurfaces can be constructed with a little work.

Now the projection from $P_i$ defines a birational isomorphism between $H$ and $\mathbb P^n$ and the composition of these defines a birational involution on $\mathbb P^n$.
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Here is an example of a birational involution of $\mathbb P^n$ that is at least not obviously a combination of Cremona transformations:

Suppose $H\subset \mathbb P^n$ is a hypersurface of degree $d$ in $\mathbb P^{n+1}$ such that it contains two points $P_1,P_2\in H$ such that the multiplicity of $H$ at $P_i$ is exactly $d-1$ for $i=1,2$. Such hypersurfaces can be constructed with a little work.

Now the projection from $P_i$ defines a birational isomorphism between $H$ and $\mathbb P^n$ and the composition of these defines a birational involution on $\mathbb P^n$.