Very arrogant, you'll notice, in shaking dice (8)
I believe the answer is:
despotic
'very arrogant' is the definition.
'despotic' can be an answer for 'arrogant' (thesaurus). I'm unsure of the 'very' bit.
'you'll notice in shaking dice' is the wordplay.
'you'll notice' becomes 'spot' (I am not sure about this - if you are sure you should believe this answer much more).
'in' is an insertion indicator.
'shaking' indicates an anagram.
'dice' with letters rearranged gives 'deic'.
'spot' placed into 'deic' is 'DESPOTIC'.
Can you help me to learn more?
(Other definitions for despotic that I've seen before include "Dictatorial" , "High-handed" , "Like a cruel ruler" , "Authoritarian" , "Autocratic" .)