You don't really need a diagonal matrix, just an upper triangular matrix.

Start with the augmented matrix

and row reduce to an upper triangular matrix. the last column will contain combinations of a, b, c, and d. Taking a=1, b= c= d= 0, then b= 1, a= c= d= 0, then c= 1, a= b= d= 0, and finally d= 1, a= b= c= 0, will give the basis vectors you need.