Can there exist a right invariant killing field of a right invariant (but not bi-invariant) Riemannian metric on a Lie group? 


I am especially interested in the case of $SU(N)$ with a metric of the form (at the identity):
$g(x,y) = \frac{1}{\lambda} B(x,y) + \frac{1}{\lambda^2} B(x,w)B(y,w)$ where $w$ is an arbitrary given vector in $\mathfrak{su}(n)$ s.t. B(w,w) < 1 and B is the Killing form (taken to be positive definite).



Right invariant Killing fields of Right invariant Riemanian metrics
A right invariant Riemannian metric is invariant und all right translations. A left invariant vector field has a flow consisting of right translations (by $\exp(tX)$).
Thus each left invariant field is a Killing field. A right invariant field $R\_X$ is Killing if and only if $S^2(\text{ad}\_X)^\*g\_{e}=0$. 


