Unfortunately I am not well versed in relativity, my response was completely rooted in linear algebra.
That is, any vector is going to be inherently one dimensional, regardless of its coordinates. Dimension is a property of a set of vectors, dependent upon how many are linearly independent of the rest of the vectors in the set. What I described doesn’t imply that there are infinite dimensions.
The vector [0,0,1] is traveling through three dimensions just the same as [1,1,1].