Lie-Algebra, Generators

[Physiana]

After reading some threads I decided to post my question, since I couldn't find an sufficient answer.
In general the generators of a lie group combined with the compostion [A,B]=ifB build a Lie-Algebra. Where the Generators build the base of the vectorspace.
In a common vectorspace you can find a orthonormalbase, and the scalarproduct defines the metrik. How does the comutation relations fit in this concept? I know that the comutation relations show if two entities can be measured together with any accuracy, but what is their meaning referring to the understanding of a algebra?
I have the feeling I disorganised the whole concept...

 

[Ratzinger]

When you transform your state vectors you use unitary operators, which are expressed as exponentials of Hermitian generators. All these operators represent together a set of space-time symmetries (rotation, displacement in time and space, velocity along an axis), which form a group. Now you can multiply these group elements, and to check if the commute you expand to small order and look how the generators behave to each other, if they commute. These form Lie Algebras.

 

[George Jones]

After reading some threads I decided to post my question, since I couldn't find an sufficient answer.
In general the generators of a lie group combined with the compostion [A,B]=ifB build a Lie-Algebra. Where the Generators build the base of the vectorspace.
In a common vectorspace you can find a orthonormalbase, and the scalarproduct defines the metrik. How does the comutation relations fit in this concept? I know that the comutation relations show if two entities can be measured together with any accuracy, but what is their meaning referring to the understanding of a algebra?
I have the feeling I disorganised the whole concept...

A Lie algebra' s Killing form is used to define a metric on the algebra. If the algebra is semi-simple, the metric is non-degenerate. If the algeba generates a compact Lie group, the metric is definite.