Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
I believe that the above calculations are correct. However, as pointed out to me by Martin Orr, if one intersects $\mathfrak{S}$ with the Levi subgroup isomorphic to $SL_2\times\mathbb{G}_m$ and then takes the image of this set in $L$ under the natural projection, then this is indeed a Siegel domain in $L$. This is essentially the last statement of the penultimate paragraph of section 4.4. Subsequently, one can choose the element $a$ from section 4.5 to be an element of the aforementioned intersection.