In this post, I will record some properties of reduced root systems, following Bourbaki. The reference is their “Lie groups and algebras, Ch. 46”. There will be no proofs, just a summary of the main properties. The post consists mainly of definitions and lists their properties.
Throughout, all root systems are assumed reduced.
Definition 1 A reduced root system is the data of a real vector space and a subset of roots such that
 (i) is finite and generates
 (ii) there exists such that
 takes to
Given , note that is unique.
 (iii) we have
 (iv) If then
Remark 2
 (i) The map fixes and takes . Therefore, gives .
 (ii) The group generated by the is called the Weyl group of , denoted .
 (iii) The coroots form the dual root system, and . Warning: The bijection is not linear.
 (iv) The root system is irreducible if it cannot be written nontrivially as .
— 1. The scalar product —
On an irreducible root system, up to scaling their is a unique scalar product on which is invariant under . Denote it by . We could take it to be, for example
If we identify with using this scalar product, then we have
Note that become orthogonal reflections in . Then acts transitively on the roots of the same length.
— 2. Possible angles —
Let . Then we have
We have the following cases:
 If then .

Then and . 
Then and . 
Then and .  but
Then and .  but
Then and .The next cases arise only for
 but
Then and .  but
Then and .
— 3. Chains of roots —
First, we have that
 If then is a root (unless )
 If then is a root (unless )
Definition 3 Given non proportional roots the set
is the chain defined by . The corresponding integer interval is . Call the origin and the end of the chain. The length is . If is the origin of an chain, then the length of the chain is .
— 4. Chambers, bases, Cartan matrix —
Definition 4 The chamber of a root system is a connected component of , where is a hyperplane.
Here are the properties:
 If then
 acts transitively on the chambers
 We have dual chambers under the linear isomorphism given by the scalar product
Definition 5 Given a chamber , a basis of is the set of roots such that form the walls of , and for all .
The properties are:
 is a basis of in the usual sense,
 Any root is an integral linear combination of elements in , having either all positive, or all negative coefficients.
 For all we have .
 Any root can be transformed to an element of using .
 The relationship with the duals is .
Definition 6 Set with the forming a basis. The matrix is called the Cartan matrix of the root system.
The properties are:
 The Cartan matrix uniquely determines .
 We have and for .
— 5. Positivity and order —
Fix a chamber and thus a basis .
Definition 7 An element is positive if is a linear combination of the with positive coefficients. An equivalent definition is that takes positive values on .
The basis can be characterized as those positive roots which are not the sum of two positive roots.
The properties are:
 Every root is either positive or negative. Let denote the positive roots.
 takes and permutes the other elements of .
 The set of positive elements in contains , but is usually bigger.
 The notion of positivity leads to a partial order relation on elements of .
 We have that if and only if for all .
 Let be a root with each positive. Then there exists a permutation such that for all we have that is also a root.
 There exists a largest root
such that for any root we have that and .
— 6. Weights (“poids” in French) —
Definition 8 The group which takes integer values for all elements of is called the weight lattice. Let be the basis dual to . These are the fundamental weights. Positive integer combinations of the are called dominant weights, denoted .
We have