Can you carry an ice axe while you are...
Can you carry an ice axe while you are carrying a snowboard?
because the snowboard carry system is on the outside and vertical, it interferes with the ice axe loops. it can be done, but you would be putting your axe under your board. its just up to you if you want expose the bottom of your board to your ice axe. with an ice axe cover it should be fine.
yes, you can do it, but think about protecting your base in some way.