Modified question There was $sl(2,\mathbb R)$ used instead of $su(2)$ in the previous version. Thanks to MattE for pointing it out.
I have seen it claimed many times that $so(4,\mathbb R)=su(2)\times su(2)$. If $C$ is a Cartan subalgebra of $so(2n)$ then the Weyl group on $so(2n)$ is generated by permutations of $(x_1,...,x_n)\in C$ and by negations of even number of $x_i$s. Hence, for $so(4)$, the orbit of $(x_1,x_2)\in C$ under the Weyl group action is $(x_1,x_2), (-x_1,-x_2), (x_2,x_1), (-x_2,-x_1).$
The Weyl group of $su(2)\times su(2)$ is the product of Weyl groups of its two components. Hence, the orbit of $(x_1,x_2)$ in a Cartan subalgebra of $su(2)\times su(2)$ under its Weyl group action is $(x_1,x_2), (-x_1,x_2), (x_1,-x_2), (-x_1,-x_2).$
That looks like a contradiction! Where do I go wrong? You can of course reformulate this question for $SO(4,\mathbb R)=SU(2)\times SU(2)$ or $SO(4,\mathbb C)=SL(2,\mathbb C)\times SL(2,\mathbb C).$
Original question
I have seen it claimed many times that $SO(4,\mathbb R)=SL(2,\mathbb R)\times SL(2,\mathbb R),$ However, the orbit of $(x_1,x_2)$ in a max torus of $SO(4)$ under the Weyl group action is $(x_1,x_2), (x_1^{-1},x_2^{-1}), (x_2,x_1), (x_2^{-1},x_1^{-1}),$ while the orbit of $(x_1,x_2)$ in a max torus of $SL(2)\times SL(2)$ under the Weyl group action is $(x_1,x_2), (x_1^{-1},x_2), (x_1,x_2^{-1}), (x_1^{-1},x_2^{-1}).$
That looks like a contradiction! Where do I go wrong?