I look for the Iwasawa decomposition of the group $SO(4,4)$?
According to some research on the net I found that their compact part $K$ of its Iwasawa decomposition $NAK$ is given by $K=SO(2)\times SO(2)$ and $\dim A=2$.
Thank you in adavance
Iwasawa decomposition of $SO(4,4)$?
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matrices
group-theory
lie-groups
lie-algebras
matrix-decomposition
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1What does ${\rm SO}(4,4)$ mean? In even dimensions there are two types of orthogonal groups over finite fields, so you need to say whether you mean ${\rm SO}^+(4,4)$ or ${\rm SO}^-(4,4)$. And referring to compact parts does not make much sense for finite groups. – 2017-02-20
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1@DerekHolt $\mathrm{SO}(4,4)$ is the subgroup of $\mathrm{SL}(8,\mathbb{R})$ which preserves the quadratic form $$Q(x_1,\ldots,x_8)=x_1^2+x_2^2+x_3^2+x_4^2-x_5^2-x_6^2-x_7^2-x_8^2.$$ – 2017-02-21
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0@DerekHolt $SO(p,q)$ is a notation for the reals, or generalizations such as real-closed fields. Over finite fields it would be somewhat a stupid notation. – 2017-02-22
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0A direct answer to your question is given here: https://mathoverflow.net/questions/163808/iwasawa-decomposition-of-the-pseudo-orthogonal-group . A root decomposition of SO(p,q) is given in Knapp: Lie Groups Beyond an Introduction, p. 315., which confirms that $K=SO(2) \times SO(2)$. – 2018-04-24