I'm self-studying tensor and index notation from McCullagh (1987) and I'm going over the definition of tensor contraction. As a concrete example, $\omega^{ij}$ and $\omega_{klm}$ are both tensors ($\omega^{ij}$ is contravariant whereas $\omega_{klm}$ is covariant). Then, with $$ \gamma_{klm}^{ij}\equiv\omega^{ij}\omega_{klm},\quad\gamma_{kl}^i\equiv\gamma_{klj}^{ij}=\omega^{ij}\omega_{klj} $$ (summation is implicit).
Then, $\gamma_{kl}^i$ is a tensor because, under transformation of variables, the transformed value is $$ \bar{\gamma}_{klm}^{ij}=\gamma_{tuv}^{rs}a_r^ia_s^jb_k^tb_l^ub_m^v\tag{$*$} $$ and, hence, summation over $m=j$ gives $$ \gamma_{kl}^i=\gamma_{tu}^ra_r^ib_k^tb_l^u.\tag{$**$} $$
(The highlighted text above is essentially verbatim from McCullagh.)
Here, I think here is how $a$ and $b$ are defined: $\omega^{ij}=f(x^i,x^j)$ and $\omega_{klm}=g(\zeta^k,\zeta^l,\zeta^m)$ and $y$ is a transformation of $x$, $\xi$ a transformation of $\zeta$. Then, $a_i^r=\partial y^r/\partial x^i$ and $b_r^i=\partial\zeta^i/\partial\xi^r$.
I can verify ($*$) without any issue but I can't verify ($**$) unless $x=\zeta$ and $y=\xi$. So my question is: is contraction only defined when $\omega^{ij}$ and $\omega_{klm}$ are both defined on the same underlying variables? If it's allowed that $x\neq \zeta$ and $y\neq\xi$, how does one show ($**$)?
Edit: strictly speaking, I don't need $x=\zeta$ and $y=\xi$ to show ($**$). What I do need is $$ \delta_i^j=a_i^rb^j_r\tag{$***$} $$ where $\delta_i^j$ is the Kronecker delta: $\delta_i^j=1$ if $i=j$ and $0$ otherwise. ($***$) is satisfied when $x=\zeta$ and $y=\xi$ but I think it might hold more generally.