Trial and Error gives:
2,1,-0.5
0,2,1
0,0,2
whose cube is the matrix you want.
"Brute strength" works. Because B is "upper triangular", you know A will also be "upper triangular" and look for a solution of the form .
Now, it's not too hard to calculate that and then that
Set that equal to B and you have 6 equations to solve for a, b, c, d, e, and f.