Can skew lines be perpendicular?

Question

In a math textbook for primary school, there's a picture like the one below. They explain which of the line segments are perpendicular and which are parallel to AD.

There's also a note stating that A1B1, D1C1, BB1 and CC1 are neither parallel nor perpendicular to AD.

I was wondering about that statement, because I learned that those four are perpendicular to AD. So I wrote to the book publisher asking about that. They replied that actually there are two conventions for this situation and they choose "not perpendicular" which, according to them, is more popular.

Is there really no consensus on that?

Edit:
I found another question here, related to a mathematical consensus: Is $0$ a natural number? and I'd accept an answer referring to some resources and clarifying whether the consensus exists or not.

Answer 1

According to Wikipedia, "skew lines are two lines that do not intersect and are not parallel."

Two sentences later it wrote "Two lines that both lie in the same plane must either cross each other or be parallel, so skew lines can exist only in three or more dimensions."

Therefore, skew lines (at least if you trust the Wikipedia definition) cannot be perpendicular.

Answer 2

I think, Since, A1B1 and AD are not on the same plane, we can't say they are parallel or perpendicular. But your title asks whether skew lines can be perpendicular or not? - I say, Yes, but if two lines are originally perpendicular and you skew them horizontally or vertically (or both) they are will not remain perpendicular.