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Suppose that $G$ is a group of matrices that is a submanifold of $M_n(\mathbb{R}) \cong \mathbb{R}^{n^2}$, with group law the usual multiplication of matrices. Let $X$ and $Y$ be two left invariant vector fields. Show that $$ [X, Y ]_{I_n} = X_{I_n}Y_{I_n} − Y_{I_n}X_{I_n}$$ where, on the right hand side of the equation, the product is just the usual multiplication of matrices.

I know that the tangent space at $I_n$ can be identified with the left invariant vector fields. But without any description of $G$ how does one show this above identity? I know $[X,Y]_{I_n}(f)=X_{I_n}Yf-Y_{I_n}Xf$ where $f$ is a smooth real valued function on $G$, but I don't see any way to connect the two above facts. How does one prove this? Thank you.

shadow10
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  • Thank you for the answer. I could solve for the case when $GL_n(R)$ but in general how does one do it? Because I felt for this case we use the structural properties of GL. But in general can we say the same thing? That's what I wanted to understand – shadow10 Mar 04 '20 at 19:56
  • By Ado's theorem every f.d. real Lie algebra can be seen as a subalgebra of $GL_n(\Bbb R)$, so that we always have $[X,Y]=XY-YX$, i.e., it is enough to understand this example. – Dietrich Burde Mar 04 '20 at 19:58
  • Oh that makes sense. But is there any way to do it directly? Without invoking such a powerful theorem? – shadow10 Mar 04 '20 at 19:59
  • Yes, we need not invoke it. It looks like we start with group law usual multiplication of matrices anyway, so we could use just the same proof as for $GL_n$. – Dietrich Burde Mar 04 '20 at 20:02

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