Definition of simple:
L is called simple if it has no ideals except {0} and L.
I is an ideal of L if [tex]x\in L, y\in I \Rightarrow [x,y]\in I [/tex]
The matrices whose trace is 0 form the special linear lie algebra [tex]sl_n(\mathbb{C})[/tex]. The special linear lie algebra is the lie algebra of the special linear group (this group is form by the matrices whose determinant is 1).