It is clear from the nature of this poem that the poem concerns the death of somebody very important to the poet. What confuses the issue is that this poem was famously used as an elegy for a lover's gay partner at his funeral in the film Four Weddings and a Funeral, and also Auden himself was gay, having long term relationships with Stephen Spender and Christopher Isherwood.
However, in spite of this, this poem actually comes from a play, entitled The Ascent of F6, which was written jointly with Christopher Ishwerwood about Michael Ransom trying to climb F6, a Himalyan mountain, to please his mother. The lament comes in the play after Michael Ransom's brother's death, James Ransom, who died. They had a very difficult relationship and Michael constantly tried to out-shine him during his life.