Pertubation of a commuting d-tuple of $2 \times 2$ matrices to a commuting *diagonalizable* d-tuple of matrices

Question

Does anybody know a simple proof of the following fact :

Any $d$-tuple of commuting $2 \times 2$ matrices can be perturbed to a $d$-tuple of commuting diagonalizable $2 \times 2$ matrices

It seems to be a very well-known fact in functional analysis, but I cannot find a simple proof of this...