Prove $T\Omega \times T x=T(\Omega \times x)$ Let $T:V\to V, T\in SO(3)$ represent a rotation matrix in $\mathbb R^3$, $\Omega \in \mathbb R^3$ is a vector, and $x\in \mathbb R^3$ also represent a vector. How to prove $T\Omega \times T x=T(\Omega \times x)$ by use of index notations?
My Attempt: Since $T\in SO(3)$, $TT^T=T^TT=1$, and we can write this as ${T^{a}}_{b}{T_{c}}^b={\delta^a}_c$. The desired result can be written as:
$$
{\varepsilon^{a}}_{bc}{T^b}_d\Omega^d {T^{c}}_e x^e={T^{a}}_{b}{\varepsilon^b}_{de}\Omega^dx^e
$$
But I have no idea how to make use of $TT^T=T^TT=1$ to get the desired result. Could anyone please help me? If I've made mistakes, please also remind me.
 A: Through the techniques @ContraKinta said in the comments, we are able to solve the problem.
Since we are going to prove
$$
{\varepsilon^{a}}_{bc}{T^b}_d\Omega^d {T^{c}}_e x^e={T^{a}}_{b}{\varepsilon^b}_{de}\Omega^dx^e
$$
Cancelling out $\Omega^d x^e$ on both sides, it is sufficient to prove
$$
{\varepsilon^{a}}_{bc}{T^b}_d {T^{c}}_e ={T^{a}}_{b}{\varepsilon^b}_{de}
$$
Multiplying both sides with $T_{ak}$, and now the task is to prove
$$
{\varepsilon^{a}}_{bc}{T^b}_d {T^{c}}_e T_{ak} ={T^{a}}_{b}{\varepsilon^b}_{de} T_{ak}
$$
For the left side of the equation:
$$
\begin{aligned}
{\varepsilon^{a}}_{bc}{T^b}_d {T^{c}}_e T_{ak} &= g^{am} \varepsilon_{mbc}{T^b}_d {T^{c}}_e T_{ak} \\
&=g^{am}  T_{ak}  \varepsilon_{mbc}{T^b}_d {T^{c}}_e \\
&={T^m}_k\varepsilon_{mbc}{T^b}_d {T^{c}}_e \\
&=\varepsilon_{mbc}{T^m}_k{T^b}_d {T^{c}}_e \\
&=\det T \varepsilon_{kde} \\
&=\varepsilon_{kde}
\end{aligned}
$$
For the right part of the equation:
$$
\begin{aligned}
{T^{a}}_{b}{\varepsilon^b}_{de} T_{ak} &= {T^{a}}_{b} g^{mb} \varepsilon_{mde} \\
&= T^{am} \varepsilon_{mde} T_{ak}\\
&= T^{am} T_{ak} \varepsilon_{mde} \\
&= g^{mi} {T^{a}}_i g_{kj}{T_a}^j  \varepsilon_{mde}\\
&= {T^{a}}_i {T_a}^j \varepsilon_{mde}\\
&= g^{mi} g_{kj}{\delta_i}^j\varepsilon_{mde} \\
&= {\delta ^m}_k \varepsilon_{mde} \\
&= \varepsilon_{kde}\\
&= \text{left side}
\end{aligned}
$$
Thus, we proved that the left and right sides are equal, which is equivalent to prove the desired equation.
