 
Summary: Journal of Algebra 287 (2005) 496500
www.elsevier.com/locate/jalgebra
Extending systems to bases of root systems
Helmer Aslaksen
, Mong Lung Lang
Department of Mathematics, National University of Singapore, Singapore 117543, Singapore
Received 13 May 2004
Available online 3 March 2005
Communicated by Georgia Benkart
Abstract
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, C, of two or more roots to a basis B of R? A system is a
linearly independent set of roots such that if and are in C, then  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 diagrams of R can
always be extended to a basis of R.
2005 Published by Elsevier Inc.
1. Introduction
Let R be an indecomposable root system in a Euclidean space V . A subset B of R is
called a basis of R if B is a vector space basis of V and each root of R can be written
