I am stuck at Problem 1.4(a) from Riemannain Geometry, written by Do Carmo.
A function $g:\mathbb R\to\mathbb R$ given by $g(t)=yt+x$, $t$,$x$,$y\in\mathbb R$, $y>0$, is called a proper affine function. The subset of all such function with respect to the usual composition law forms a Lie group $G$. As a differentiable manifold $G$ is simply the upper half-plane $\{(x,y)\in\mathbb R^2|y>0\}$ with the differentiable structure induced from $\mathbb R^2$. Prove that:
The left-invariant Riemannian metric of $G$ which at the neutral element $e=(0,1)$ coincides with the Euclidean metric ($g_{11}=g_{22}=1,g_{12}=0$) is given by $g_{11}=g_{22}=\frac{1}{y^2},g_{12}=0$.
I calculate the left-invariant Riemannian metric from its definition $$g_{11}=\left\langle\frac{\partial}{\partial x}|_{(x,y)},\frac{\partial}{\partial x}|_{(x,y)}\right\rangle_{(x,y)}=\left\langle(L_{(x,y)}^{-1})_*(\frac{\partial}{\partial x}|_{(x,y)}),(L_{(x,y)}^{-1})_*(\frac{\partial}{\partial x}|_{(x,y)})\right\rangle_e$$
Now $G$ is a half-plane with the composition law: $(x_1,y_1)(x_2,y_2)=(x_2+x_1y_2,y_1y_2)$. So $$L_{(x,y)}^{-1}(a,b)=\left(a-\frac{x}{y}b,\frac{1}{y}b\right)$$
Then $g_{11}=\left\langle\frac{\partial}{\partial x}|_e,\frac{\partial}{\partial x}|_e\right\rangle_e=1$
This isn't equal to $\frac{1}{y^2}$. Where do I make mistakes? Any advice is helpful. Thank you.