Undergrad On the Coriolis Forcing vector and its Matrix

[Gear300]

The context for the question is in the attachments (pg1.png, pg2.png, pg3.png), so there is some reading involved. Although, it is a short and simple read if anything. The inquiry is in (inquiry.png).
My understanding of the situation is that Q(t) abides by the differential equation
Q'(t)Q(t)^T + Q(t)Q'(t)^T = 0 .
So by something akin to Picard's existence theorem on differential equations, we can resolve exact trajectories for Q(t) given initial conditions. The peculiar thing is that
Q(t₀) z(t₀) = z(t₀)
can likely be assumed to be true for all z(t₀) ∈ ℝⁿ, in which case
Q(t₀) = I
should also be true.
Now if I let R = ( i(t₀) , j(t₀) , k(t₀) ), then
R^TR = RR^T = I ,
and the question insists that
R^TQ(t)R = Q ⇒ Q(t)R = RQ(t).
Although, our choice of the initial condition basis vectors for R is arbitrary, so how can we expect that R necessarily commutes with Q(t)? Unless of course, the question is asking for the matrix expression of Q(t) in the { i(t₀) , j(t₀) , k(t₀) } basis. I am just wondering if I am to expect the matrix expression of Q(t) to remain invariant given any orthonormal basis of the same orientation?
(Also, how do I insert latex?)

 

[Gear300]

Well, by the looks of it, the matrix expression for Q(t) would change depending on the basis { i(t₀) , j(t₀) , k(t₀) } used. So I suppose the answer is simply in the context of the trajectory of one point.