Jumping Jack Flash scale

Trying to work this one out.
From what I can gather it is in B Mixolydian, but the chords are A, B, D, E, which as far as I know do not fit any scale.
Apart from by ear, how do I know what notes to pick for each individual chord?

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Welcome back, Brian!

Mixolydian has a flat 7th scale degree.

B Mixolydian is therefore:

B, C#, D#, E, F#, G#, A, B

Harmonised it gives these chords:

B → (or B7) → I
C#m → (or C#m7) → ii
D#dim → (or D#m7b5) → iii
E → (or Emaj7) → IV
F#m → (or F#m7) → v
G#m → (or G#m7) → vi
A → (or Amaj7) → bVII

Note, these are the same chords you would find in the key of E major. Look at it series, B Mixolydian is the 5th mode of E major (Ionian).

You have the I bouncing down to the bVII and back again:

B → A → B → A etc.

In the refrain you have bIII (a borrowed chord) to bVII to IV to I:

D → A → E → B repeating

If you consider that B Mixolydian has E major as its parent major scale, the D chord = bIII is borrowed from the parallel key of E Minor (Aeolian).

More here:

And here: