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To rotate your object on a specific local axis, you can do the following:

• First, convert your object into an instance (if it does not already exist as an instance).
• Then rotate the instance with Rotate Instances time-controlled with Scene Time on the Z-axis in the local space.
• Then rotate the instance in global space by the value $22.5°$.

Note: When you plug a value into the socket Rotation, the node expects the rotation in radians, and not in degrees!

PS: Of course you could also work with the node Transform, but for me the example using the instances seemed easier, because you can easily select the local or global space.

_{(Blender 3.1+)}

Why does this not work as expected in your example?

Your screenshot shows a particularly unfavorable situation:

• First you multiply the seconds by $2$. This is a float value. So let's assume for second $1$ that this value is then $2$.
• Then you put this float value into a vector input, which automatically leads to the value $(2,2,2)$. But since the input is a Rotation, and angles in radians are expected here, this means a rotation of $114.598°$ per axis per second.
• In addition, Rotate Euler does not actually work here, because no rotation axis has been specified.
• Then, by putting the rotation into a float input of Combine XYZ, you convert it back into a float value. And in doing so, the value is divided by $3$ again.

That's the reason why you don't get a usable rotation at the end here.