Let $G$ be a Lie group and $H$ a closed subgroup. We denote by $mathfrak g$ and $mathfrak h$ by the Lie algebras of $G$ and $H$ respectively. For $xin G,$ denote $tau(x):G/Hto G/H$ by $gHmapsto xgH.$ Denote $pi:Gto G/H$ to be the quotient map and put $o=pi(e)$ where $e$ is the identity of $G.$ For $hin H$ denote $dtau(h):=dtau(h)(o)$ and $dpi:=dpi(e).$ I want to understand the following Lemma (lem 1.7) of Helgason’s book `Groups, Geometry and Analysis’

$det(dtau(h))=frac{det Ad_G(h)}{det Ad_H(h)}.$

Helgason proves that if $mathfrak m$ is any subspace so that $mathfrak g=mathfrak hoplus mathfrak m$, then for any $hin H$ we have $dpi|_{mathfrak m}circ A_h=dtau(h)circ dpi|_{mathfrak m},$ where $A_h(X)$ is the projection of $A_G(h)X$ onto $mathfrak m$ for $Xinmathfrak m.$ Therefore, $det A_h=det(dpi(h))$. Also $Ad_G(h)X=Ad_H(h)X$ for $Xinmathfrak h.$ I understand upto his point. Now Helgason directly draws the conclusion that $det Ad_G(h)=det(dtau(h))det(Ad_H(h)).$ I do not understand this. It seems that the problem will be solved if we can show that $Ad_G(h)=Ad_H(h)oplus A_h$. But this seems unlikely.