Orthogonal Projections in Lie Theory ?

I have been studying a finite element method where rigid & elastic spatial motions are separated using an orthogonal projection (actually two: one for translations/stretches, the other for rotations/spins). Since all rigid & elastic modes are orthogonal with respect to the standard Euclidean inner product, I understand the projection's action in terms of linear algebra. However, I'd like to think about it from a Lie theory perspective. So far, I haven't found any literature discussing this topic. I'm wondering if the topic makes any sense &, if so, where I might explore it. Thanks, John.

After more reading, I have found the following which may help:

i) If I and J are two ideals in a Lie algebra g with zero intersection, then I
and J are orthogonal subspaces with respect to the Killing form.

ii) If a subspace I of Lie algebra g satisfies a stronger condition that[g, I]
is a subspace of I, then I is called an ideal in the Lie algebra g.

iii) The Killing form for R^3 is the standard Euclidean inner
product.

If this is the right idea, then I need to show that rigid motion
(e.g. I) & elastic deformation (e.g. J) are ideals with zero
intersection.

Thanks, John.