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I am sorry That I am getting involved so late. I was away at a meeting for two days. I have an old (1961) paper in the ANNALS OF MATH called "On the Existence of Slices for Actions of Non-Compact Lie Groups" which is quite relevant. In particular, on page 318 you can find the following concerning proper actions of an arbitrary Lie group $G$.

Theorem 4.3.4. Every seperable, metrizeable, proper $G$-space $X$ admits an invariant metric. ...

There are a large number of other theorems there that show that the theory of proper G-spaces for G an arbitrary Lie group is similar to the theory of G-spaces for a compact Lie group. If you have access to JSTOR you You can find the paper here: http://www.jstor.org/pss/1970335 If you do not have access to JSTOR and would like a copy, let me know (palais@uci.edu) and I will arrange to send you a copy.http://vmm.math.uci.edu/ExistenceOfSlices.pdf
show/hide this revision's text 1

I am sorry That I am getting involved so late. I was away at a meeting for two days. I have an old (1961) paper in the ANNALS OF MATH called "On the Existence of Slices for Actions of Non-Compact Lie Groups" which is quite relevant. In particular, on page 318 you can find the following concerning proper actions of an arbitrary Lie group $G$.

Theorem 4.3.4. Every seperable, metrizeable, proper $G$-space $X$ admits an invariant metric. ...

There are a large number of other theorems there that show that the theory of proper G-spaces for G an arbitrary Lie group is similar to the theory of G-spaces for a compact Lie group. If you have access to JSTOR you can find the paper here: http://www.jstor.org/pss/1970335 If you do not have access to JSTOR and would like a copy, let me know (palais@uci.edu) and I will arrange to send you a copy.