For a pair of $(X,Y)$ of linearly independent vectors in $T_pM$, $p\in M$, the sectional curvature is defined as
$$K_p(X,Y)=\frac{<R(X,Y)Y,X>}{|X|^2 |Y|^2 - <X,Y>^2}$$
The problem I'm looking at now asks us to compute the same on the half-plane $\mathbb{H} = \{(x,y) \in \mathbb{R}|y>0\}$ with the metric $g= 1/y^2(dx^2 + dy^2)$ (and hence show that it is the constant $-1$). The provided solution has an equality that I don't understand:
$$K(\partial_1, \partial_2) = \frac{<R(\partial_1,\partial_2)\partial_2,\partial_1>}{(1/y^2)(1/y^2) - 0} = y^4\frac{1}{y^2}R^2_{112}$$
How is this last equality justified? Namely, I want to understand how the author arrived at this particular set of indices for $R$. I see that the metric is involved because of the inner product, but the details escape me. A breakdown of the steps would be much appreciated!
Note that $E_1 =\partial_x,\ E_2=\partial_y$ are coordinate vector field From this $e_1=y\partial_x,\ e_2=y\partial_y$ are orthonormal vector fields.
$${\rm sec}\ (e_1,e_2)=R(e_1,e_2,e_2,e_1)=\frac{R(E_1,E_2,E_2,E_1)}{ (E_1,E_1)(E_2,E_2)-(E_1,E_2)^2}$$ $$ = \frac{ R_{1221}}{ 1/y^4} = y^4 R_{122}^k g_{k1} =y^2 R_{122}^1 $$
(Here index is wrt coordinate vector field)