I'm a bit skeptical as to how simple and low-tech a proof can be, since a fair amount of Lie group theory has to be in hand.   Short of going into the full details of Iwasawa decomposition, you might find it worthwhile to look at two old papers in the Bulletin of the AMS (freely available online).  One is by <a href="http://www.ams.org/journals/bull/1949-55-10/S0002-9904-1949-09325-4/home.html">Mostow</a>, giving a new proof of Cartan's theorem, while the other is a longer survey on the topology of Lie groups by <a href="http://www.ams.org/journals/bull/1952-58-01/S0002-9904-1952-09544-6/home.html">Samelson</a>.