Let $G\subset GL(n,\mathbb{R})$ be a connected linear Lie group. Asumme that $[\frak{g},\frak{g}] = \frak{g}$. Show that $G\subset SL(n,\mathbb{R})$.

Question

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Let $G\subset GL(n,\mathbb{R})$ be a connected linear Lie group, and $ \frak{g}$$ = Lie(G)\subset M(n,\mathbb{R})$. Asumme that
$[\frak{g},\frak{g}] = \frak{g}$. Show that $G\subset SL(n,\mathbb{R})$.

Answer 1

$[{\cal g},{\cal g}]={\cal g}$ implies that if $x\in {\cal g}$, $x=\sum [u_i,v_i], u_i,v_i\in {\cal g}$. This implies that $tr(x)=0$, ${\cal g}\subset sl(n,\mathbb{R})$ and $G\subset Sl(n,\mathbb{R})$.