BiMagic

Consider the first row of the order 16 composite:

Let A0 be aspect 0 of the 2x2 square, A2 be aspect 2 of the 2x2 square,
    B be the 8x8 square, and C be the 16x16 composite, and
let their first row elements be (a00, a01),(a20, a21), (b0 .. b7), (c0 .. c15).
Also, take B as 0-based, first row (2 20 50 13 36 27 61 43).

then:
c02+c12+c22+c32+c42+c52+c62+c72+c82+c92+c102+c112+c122+c132+c142+c152 =

(a00+22b0)2 + ... + (a01+22b7)2 =

(a002+8a00b0+16b02) + (a012+8a01b0+16b02) + (a002+8a00b1+16b12) + (a012+8a01b1+16b12) +
(a202+8a20b2+16b22) + (a212+8a21b2+16b22) + (a202+8a20b3+16b32) + (a212+8a21b3+16b32) +
(a202+8a20b4+16b42) + (a212+8a21b4+16b42) + (a202+8a20b5+16b52) + (a212+8a21b5+16b52) +
(a002+8a00b6+16b62) + (a012+8a01b6+16b62) + (a002+8a00b7+16b72) + (a012+8a01b7+16b72) =

  I: 4(a002+a012+a202+a212) +

 II: 8( (a00+a01)(b0+b1+b6+b7) + (a20+a21)(b2+b3+b4+b5) ) +

III: 2×16(b02+b12+b22+b32+b42+b52+b62+b72)

-------------------------------
let S2   be the order  2   magic sum =  2(22+1)/2     = 5
    S8   be the order  8   magic sum =  8(82-1)/2     = 252
    S16  be the order 16   magic sum = 16(162+1)/2    = 2056
    S22  be the order  2 bimagic sum = S2(2×22+1)/3   = 15
    S28  be the order  8 bimagic sum = S8(2×82-1)/3   = 10,668
    S216 be the order 16 bimagic sum = S16(2×162+1)/3 = 351,576
if the 2x2 squares were bimagic, the formula would be:

   S216  = 8(S22) + 8(S2)(S8) + 32(S28)
 351,576 = 8(15)  + 8(5)(252) + 32(10,668)
-------------------------------
we have:
III: 2×16(b02+b12+b22+b32+b42+b52+b62+b72) = 32(S28)

so, barring a combination of I and II, we need:
 I: a002+a012+a202+a212 = 2(S22) = 2(15) = 30
and, we have:
    12 + 32 + 42 + 22  = 1 + 9 + 16 + 4 = 30

and, we need:
II: (a00+a01)(b0+b1+b6+b7) + (a20+a21)(b2+b3+b4+b5) = (S2)(S8) = (5)(252) = 1260
and, we have:
        (1+3)(2+20+61+43) +     (4+2)(50+13+36+27) = (4)(126) + (6)(126) = 1260

Note that the order 8 magic sum has two sub-sums of 126 as needed for 
the two 2x2 aspect sums 1+3 and 4+2.

No BiMagic

No Pattern

There are no 2x2 square 2b or 2c aspect patterns that result in a bimagic composite sum with the main back diagonal, (1 53 6 42 27 46 50 35), of this 8x8 bimagic square: .txt

A component of the bimagic formula is:

                          8(S2)(S8) = 8(5(252))

(for 0-based numbers in the 8x8 square).

The 2x2 2b diagonal sums are 3 and 7. There is no partition (x,y) of the diagonal numbers,
(0 52 5 41 26 45 49 34), for which:

                      3(sum(x)) + 7(sum(y)) = 5(252)

Similarly, the 2x2 2c diagonal sums are 4 and 6, and there is no partition for which:

                      4(sum(x)) + 6(sum(y)) = 5(252)

No Combination

There are 2x2 square 2a aspect patterns that result in a bimagic composite sum with each row, column, and main diagonal of the above 8x8 bimagic square, but these patterns do not combine to make any 8x8 pattern.