Extending -systems to bases of root systems

Let R be an indecomposable root system. It is well known that any root is part of a basis B of R. But when can you extend a set of two or more roots to a basis B of R? A -system is a linearly independent set of roots, C , such that if ̨ and ˇ are in C , then ̨ ˇ is not a root. We will use results of Dynkin and Bourbaki to show that with two exceptions, A3 Bn and A7 E8, an indecomposable -system whose Dynkin diagram is a subdiagram of the Dynkin diagram of R can always be extended to a basis of R.