I believe this is true in any building (which IG is). Equivalently, this is one of the usual properties of BN-pairs. The way you can think about it is that I is generated by $Icap w_2 I w_2^{-1}$ and $Icap w_1^{-1}Uw_1$ (since no positive root space can be sent to a negative one by both $w_2$ and $w_1^{-1}$).
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