I would like an explicit description of $\mathbb{R} SO(n) I\_n$, i.e., the image of the identity under the action of the group algebra of $SO(n)$ by left multiplication. Equivalently, what is an explicit description of the vector space $\{\sum\_i c\_i A\_i: A\_i \in SO(n)\}$? Presumably this is well-known and in the context of all compact simple Lie groups? I need this to understand some continuous state space Markov chain of particle systems. 


Edit: to those who vote to close, please state reason. If you have a one-liner answer, why not give it a shot? I will close it myself when I see a satisfactory response. 



Orbit of the identity matrix under Lie group algebra actions
For $n\geq3$, it is the full space of $n\times n$ real matrices. The reason is that 
the $SO(n)$-orbit through the identity is the same as the $SO(n)\otimes SO(n)$-orbit by left
and right multiplication, $(g,h)\cdot X=gXh^{-1}$, and this representation is irreducible for $n\geq3$. (Note that the span of any orbit is an invariant subspace, in general). 


