Is killing form a norm?

Question

This referees to the link http://mathworld.wolfram.com/KillingForm.html , where it clearly states that killing form is an inner product. Further it is also known that inner product induces a norm. So does killing form induces a norm?

EDIT (After viewing some answers): Some excellent answers proved that killing form cannot be norm in general. At least is there a condition under such the killing form can be a norm?

Answer 1

In general, the killing form is not necessarily an inner product. If the killing form is degenerate, then it is not an inner product. Also, Lie algebras can be over any field while inner products map to real of complex numbers

Answer 2

The killing for is non degenerated if and only if the Lie algebra is semi-simple. It is an inner product if the Lie algebra is compact and semi-simple in this case it is negative definite.

https://en.wikipedia.org/wiki/Compact_Lie_algebra