I'm self studying the book Cartan for beginners and i'm trying to solve the exercise 2.5.2. The setup is the following:
Let $\operatorname{ASO}(n+s)$ the set of rototranslation in the $n+s$ dimensional affine space $\mathbb{E}^{n+s}=\operatorname{ASO}(n+s)/\operatorname{SO}(n+s)$ and let $M$ an $n$-dimensional submanifold.
The space $\operatorname{ASO}(n+s)$ can be identify with frame bundle of the affine space i.e. we have $\pi : \operatorname{ASO}(n+s) \rightarrow \mathbb{E}^{n+s}$. Let $\pi :\mathcal{F}^1 \rightarrow M$ the subbundle of whose fibers over $x \in M$ are the orthonormal frames $(x,e_1,\cdots, e_{n+s})$ such that $e_1, \cdots , e_n$ are tangent to $M$ and let $$ \omega=\begin{pmatrix} 0 & 0 & 0\\ \omega^i & \omega^i_j & \omega^i_b \\ \omega^a & \omega^a_j & \omega^a_b \end{pmatrix} $$ $0\le i,j\le n$, $n+1\le a,b \le n+s$ the Maurier Cartan form of $\operatorname{ASO(n+s}$. The exercise asks to show that $$ \tilde{I}:=\sum_i \omega^i \omega^i \in \Gamma(\mathcal{F}^1, \pi^*(S^2 T^*M)) $$ descends to a well defined differential invariant in $\Gamma(M,S^2T^*M)$ (in particular it should be the first fundamental form) i.e. that given two sections $s_1,s_2 : M \rightarrow \mathcal{F}^1$ we have $s_1^*(\tilde{I})=s_2^*(\tilde{I})$.
I know that,given two section of $\mathcal{F}^1$ we have that $s_2=s_1R$ where $$ R=\begin{pmatrix} 1&0&0\\ 0& g^i_j&0\\ 0& 0 & u^a_b\\ \end{pmatrix} $$ where $g$ and $u$ are rotation matrices. So i obtain $$s_2^*(\tilde{I})=\sum_i (g^{-1})^i_j (g^{-1})^i_j s_1^*(\omega^j) s_1^*(\omega^j)$$ i think i have missed a transpose in this formula, but i can,t understand what i did wrong