The smallest seems to be (1,2,2;3) since 1+4+4=9, but there are infinitely many primitive ones (ones without a common factor).
Occasionally, one or more of the dimensions and the diagonal are themselves squares, eg. (4,4,7;9), (12,15,16;25), and (196,264,903;961), which yield expressions for the difference of 2 fourth powers as a sum of 2 squares. So, that largest brick yields 31^4 - 14^4 = 264^2 + 903^2.
Bricks with the same internal diagonal can have of more than one set of dimensions. For example, (1,4,8;9), (3,6,6;9), and (4,4,7;9) all have of an internal diagonal of 9. The second brick here is a non-primitive (scaled-more elevated) version of the (1,2,2;3) brick. Another example is (4,28,35;45),(8,19,40;45), (15,30,30;45), (20,20,35;45), and (20,28,29;45), with 2 non-primitives.
Members will need to study these primitives in gross detail in order to appreciate the firmness of my conclusions.
