As the article does not mention the jurisdiction, I suppose it’s  the Kingdom of Saudi Arabia. (1)

Why does Algosaibi’s bank not sue Algosaibi in the land of the dollar regime?

No, only Algosaibi (and Bahrain’s ABC) is (are) allowed to sue Saad Group in the said land of the free (so-called land, forcing dollar pegs). (2)

Ivo Cerckel
honestmoney@maktoob.com
http://twitter.com/ivocerckel/

(1)
Mashreq confirms it is suing Algosaibi
Dubai: 1 hour and 31 minutes ago http://www.tradearabia.com/news/newsdetails.asp?Sn=BANK&artid=164715
SNIPS
Dubai-based Mashreq bank confirmed on Tuesday it is suing Ahmad Hamad Algosaibi & Bros (AHAB), one of two Saudi conglomerates involved in a large financial dispute, over alleged irregularities.
+
The Algosaibi lawsuit said Mashreq sued it for $150 million that it does not owe.

(2)
lawsuit vs Saad Group in London demanding dollars
Posted by Ivo Cerckel on July 20th, 2009
http://bphouse.com/honest_money/2009/07/20/lawsuit-vs-saad-group-in-london-demanding-dollars/
[If the link does not work try to copy it and to paste it in your browser]