NKTs? (Natural Killer T cells)

Here's a practical question. I want to clone out human CD8+ T cells, starting with frozen PBMCs. Before I stimulated them with CMV peptide (and an artificial APC), I used magnetic beads to produce a somewhat more purified population. I used anti-CD56 beads to remove NK cells, and anti-CD19 beads to remove B cells. After a few rounds of stimulation/proliferation followed by rest, I sorted only the CD8+ cells also expressing early activation markers (CD69 and CD137). I then cultured these through several more cycles of stimulation and rest.

I then stained the cells with certain markers I was interested in, and performed flow cytometry. I found 14% of the population of one of the cultures to be CD8+, CD4+, CD3+ triple positive cells. Unfortunately, I did not stain with anti-CD56. My guess is that these peripheral double positive T cells are NKT cells. Does anyone have any ideas?