UPDATE__ Suppose l is 7 mod 8; consider the vectors W in Z^3 with(W,W)=2l. There is a group of order 48 operating on the set of such W by permutation and sign change of co-ordinates; the group operates without fixed points. So if there are 12h such W there are h/4 orbits under the group action.
----Ira Gessel's calculations, carried out for l<1500, indicate that there is an involution, O-->O' on the set of orbits, which has the following property. Let O be any of the (h/4) orbits and (r1,r2,r3) be a representative of O with r1 even (so that r2 and r3 are odd).Then if (s1,s2,s3) is a representative of O', we have the explicit identity C((r1)/2,(r1)/2,(r2+r3)/2,(r2-r3)/2)+C(s1,s2,s3)=C.
----But to know what these conjectured(but true beyond possibility of doubt) equations are for l>1500, we need to describe the involution. Franz Lemmermeyer suggested that the involution comes from an involution on a set of equivalence classes of quadratic forms ofdiscriminant -8l. This is surely the case; I'll explain what the involution on the forms is, and how to transfer it to the orbits.
----Consider positive quadratic forms rx^2+2sxy+ty^2 with s^2-rt=-2l. Gauss showed that these fall in exactly h equivalence classes under the action of SL_2(Z), where 12h is the number of W with (W,W)=2l; we'll be interested in GL_2 equivalence however. Since rt=2l+s^2,we find that mod 16, rt is 2,7,14 or 15. This can be used to show that one of the following possibilities must occur:
a.--- Every non-zero n represented by the form is the product of an integer that is 1 or 7 mod 8 by a power of 2.
b.---Every non-zero n represented by the form is the product of an integer that is 3 or 5 mod 8 by a power of 2.
----In the first case we say that the form is in the principal genus, while in the second that it is in the non-principal genus. There are (h/4) GL_2 classes in the non-principal genus. Furthermore there is an involution on this set of classes taking the class of rx^2+2sxy+2ty^2 to the class of 2rx^2+2sxy+ty^2. I'll call this involution "composition with 2x^2+ly^2".
----I now describe a map from the set of (h/4) orbits to the set of (h/4) classes. The map can be shown to be onto, and so is bijective. When we transfer composition with 2x^2+ly^2 to the set of orbits, we get our desired involution; one which is in complete accord with Gessel's calculations. Suppose (W,W)=2l. Let W# consist of all elements of Z^3 orthogonal to W. We attach to W the class of the form (xU+yV,xU+yV), where U and V are a basis of W#. This class is evidently independent of the choice of basis; one can show that it consists of forms of discriminant -8l and lies in the non-principal genus. This gives the desired map from orbits to classes of forms; as I've indicated it is bijective.
EXAMPLE____Take l=1567, and W=(3,25,50) so that (W,W)=2l. Let O be the orbit of W. I'll calculate O', and write down the conjectured equations coming from O and O'. A basis for W# consists of U=(0,2,-1) and V=(25,1,-2). Then (U,U)=5, (U,V)=4, (V,V)=630, and a form attached to O is 5x^2+8xy+630y^2. Composition with 2x^2+1567y^2 takes this to 10x^2+8xy+315y^2. So we seek U' and V' with (U',U')=10, (U',V')=4, and (V',V')=315. Take U'=(3,1,0). A little experimenting, writing 315 as a sum of 3 squares, shows that we should take V'=(5,-11,13). Then W' which is orthogonal to U' and V' can be taken to be their vector product (13,-39,-38). So O' is the orbit of (13,38,39). And one of our predicted expressions for C is C(25,25,11,14)+C(13,38,39), while another is C(19,19,13,26)+C(3,25,50).