Let $A=\frac{a}{d}, B=\frac{b}{d}, C=\frac{c}{d},$ then we get  

$$A^4+B^4+C^4=2\tag{1}$$

Let $A=x+y, B=x-y, z=C^2$ then 

$$2x^4+12y^2x^2+2y^4+z^2 = 2\tag{2}$$

Hence  

$$y^2 = -3x^2 \pm \frac{\sqrt{32x^4+4-2z^2}}{2}\tag{3}$$

So we find the rational solutions of $(4)$.  

$$v^2=32x^4+4-2z^2\tag{4}$$ 

$(4)$ can be parameterized to $(5)$ using $(z,v)=(4x^2,2)$ with $w=(k^2+2)C$.

$$w^2 = 4(k^2+2)(-2+k^2)x^2-4(k^2+2)k\tag{5}$$

Thus we must find the rational solutions of simultaneous equations $(3),(5)$.  

First we get a parametric solution of $(5)$ using giving some rational number $k$. 


Take $k=\frac{-9}{13}$ the we get 

$$x = \frac{3}{142}\frac{28561m^2-430732-7081100m}{28561m^2+430732}$$
$$w = \frac{-1257}{11999}\frac{-10768300+173732m+714025m^2}{28561m^2+430732}$$
$$C = \frac{-3}{71}\frac{-10768300+173732m+714025m^2}{28561m^2+430732}$$

Hence $(3)$ becomes to 
 
$$y^2 = \frac{10083247442281m^4-41255619857608m^2+2788930240200m^3+2293337020040464-42060204482400m}{20164(28561m^2+430732)^2}$$

Hence we must find the rational solutions $(m,V)$ of $(6)$  
$$V^2 = 10083247442281m^4+2788930240200m^3-41255619857608m^2-42060204482400m+2293337020040464\tag{6}$$

Quartic equation $(6)$ is birationally equivalent to the elliptic curve using a known solution $(m,V)=(360/169,-47445892)$ as follows. 
$$Y^2 = X^3 -X^2 -1097465452X+ 3288951361780\tag{7}$$  

Though I couldn't find the generator of elliptic curve , I got one solution 
 $P(X,Y)=(\cfrac{-97636631990}{5536609}, \cfrac{53963430434179560}{13027640977})$.  

Since $(7)$ has rank $1$, we get infinitely many solutions of $(7)$ using group law as follows.  

For instance, we get a solution of $(6)$,  
 $2Q(m,V)=(\cfrac{-762617488871059540}{36474714629699307},\cfrac{-63980049963485293932709632365019549028}{46581170383319934672021751209})$  

We get $(x,y)=(\cfrac{4317501713916820330332746705607}{16486256234272723829845043984402},\cfrac{12346241358503326020929860043561}{16486256234272723829845043984402})$  

Finally, we get 

   $a = 4014369822293252845298556668977$  
   $b = 8028813922964684804294250901215$  
   $c = 8331871536210073175631303374584$  
   $d = 8243128117136361914922521992201$