I have a Stata program, metricp, which takes a distance matrix (square, symmetric, zero diagonal) and tests for the triangle inequality (that for all i, j, there is no k such that d[i,j] > d[i,k] + d[j,k]).
I needed something similar today in R, and found Matthew Vavrek’s fossil package, which includes a tri.ineq() function. However, it turned out to be very slow on the large matrix I threw at it, so I decided to speed it up with some of the techniques in metricp.ado.