I wasnt quite sure where to put this thread. This question occurred to me while I was looking at the group theory of standard model groups so I thought I'd put it here. Anyway, here is my question: One can define the Killing form for a group by taking the trace of two generators. One can then choose a basis in which this quantity is diagonal and has entries + or - one. This appears to have the same form as a metric, and in fact in my notes the same symbol is used to denote the metric and the killing form. They both have many similar properties. But it doesnt seem to me like there should be any connection between the two - the killing form is a feature of the group, whereas the metric is a feature of the space we are working in. Are they indeed the same thing, and if they are, then why?