I've just started learning Lie algebra and got confused by the A1xA1 root system, which consists of the roots α, β, -α, and -β with <α,β>=0 so α and β form a right angle. There are 2 generators for this algebra, e1 and e2 and the algebra is of infinite dimension since the [e1,e2], [e1,[e1,e2]], ....are all independent. My question is if it's infinite dimensioned, shouldn't there be other roots corresponding to vectors other than e1, e2? I did some simple calculations which I'm sure are wrong somewhere but can't figure out where.
adh1([e1,e2])=[h1,[e1,e2]]=-[e2,[h1,e1]]-[e1,[e2,h1]]=2[e1,e2]-0=2[e1,e2]. Same with h2. So it seems that [e1,e2] is also a root space with the root=α+β, which is contradictory to the root system structure. Anyone has any ideas? Thanks!