Let   be a list of roots for either an abstract root system or for a simple Lie algebra.  In particular, must have an even number of elements and if then .  Write  where, if  then and then The set is called the set of positive roots. The choice of positive roots is not unique.

 If   is set of positive roots,then a root is called a simple root if it is not a sum of any other 2 positive roots.

If  is a set of simple roots for , then every root in is a linear combination of the roots in  with positive integer coefficients.

The number of simple roots equals the rank of the Lie algebra.