Obtaining Inverse configuration space metric $M^{AB}$ in canonical gravity

I am considering phase space of configuration variables in general scalar tensor theory $(q^A=(h_{ab},phi))$, characterised by the metric:

$ M_{AB}=frac{sqrt{h}}{N}begin{pmatrix} frac{U(phi)}{2}G^{abcd}&-U'(phi)h^{ab}-U'(phi)h^{cd}&G(phi)end{pmatrix};
$

prime denotes derivative w.r.t scalar field $phi$) and $h$ is standard spatial metric in ADM decomposition.

From relation:

$M_{AC}M^{CB}=delta_A^B,;; delta_A^B=diag(delta_{ab}^{cd},1)$
where $delta_{ab}^{cd}=G_{abkl}G^{klcd}=delta_{(a}^c delta^{d}_{b)}.$

Supermetric is defined by $ G^{abcd}=frac{1}{2}(h^{ac}h^{db}+h^{ad}h^{cb}-2h^{ab}h^{cd})$, $G_{abcd}=h_{a(c}h_{d)b}-frac{1}{2}h_{ab}h_{cd}).$

The inverse metric on the configurational space is given by:

$ M^{AB}=frac{N}{sqrt{h}}begin{pmatrix} frac{2}{U(phi)}G_{abcd}+s(U’/U)^2h_{ab}h_{cd}&-s(U’/U)h_{ab}-s(U’/U)h_{cd}&send{pmatrix}$
with $s=frac{U}{(GU+3(U’)^2)}$

I have trouble with obtaining this inverted metric. While standard metric inverse satisfy $g g^{-1}=I$, applying similar procedure led me to set of 4 equations for 4 components, i.e $M^{AB}$‘s. However, "brainless" computation didn’t worked, since one deals with tensorial quantities such as $h$‘s and supermetric $G^{abcd}$ or double delta symbol (in $(1,1)$-th component). For example, using naive substitutions i have for $M^{22}$ component expression $frac{G^{abcd}(U/2)}{G^{abcd}(U/2)-(U’)^2h^{cd}h^{ab}}$ (correct one is $s$)

As a reference, im following section 6.2 (eqs. (6.11), (6.15)) of https://inspirehep.net/literature/1784294
(in this ref., metric $h=gamma$ and $U_1=U’$).

How to perform this inversion correctly in order to get $M^{AB}$?