I guess the reason for so small socket is the desire to save space in the lantern and the short expected burning time (only the restrike time of the main HID source, so few minutes), so the thermal inertia would keep the temperature at reasonable level.

I have one of these myself, but with a E11 cap, to this day I have never found a Socket/Lampholder for them for lamp see HERE

All the Best

Colin

Mine fit fine in a SES socket, though I was not going to light them in a plastic socket only rated for a 15W lamp, and with a 1A current limit too.

These are halogen intermediate lighting lamps, that are connected to the ballast of the HID lamps, and turns on when the HID lamp, shuts down because of a power interuption or cycling.
Usually used in hi/low bay fixtures.
Double ended halogen lamps of the same wattage are more common in Israel for intermediate lighting of HID fixtures then screwbased lamps like these.