LG Ultrafine 4K Display with Thunderbolt 3 Use with Thunderbolt 2?

It should work with a Thunderbolt 3 adapter.

Thunderbolt 3 (USB-C) to Thunderbolt 2 Adapter - Apple

"In addition, it can be used to connect Thunderbolt-enabled displays — such as the Apple Thunderbolt Display and LG Thunderbolt 2 displays — to any of the Thunderbolt 3 (USB-C) / USB 4 ports on your Mac."

It should work in the other direction, too.

"As a bidirectional adapter, it can also connect new Thunderbolt 3 devices to a Mac with a Thunderbolt or Thunderbolt 2 port and macOS Sierra or later."

Be sure to use a Thunderbolt cable (Lightning bolt icon on plug). The LG display comes with two cable: one USB-C only, the other Thunderbolt 3.

Correction. You don't use the cable that came with the display. The Thunderbolt 3 adapter plugs into one of the display's Thunderbolt 3 ports (Lightning biolt icon next to the port) and you use a Thunderbolt 2 cable from the adapter to a Thunderbolt port on the computer.

Apple Thunderbolt Cable (0.5 m) - White - Apple

That’s not correct I’m afraid, you’ll need to ask LG:

“Note: Products sold through this website that do not bear the Apple brand name are serviced and supported exclusively by their manufacturers in accordance with terms and conditions packaged with the products. Apple’s Limited Warranty does not apply to products that are not Apple branded, even if packaged or sold with Apple products. Please contact the manufacturer directly for technical support and customer service.”

Ask the MacBook community? This one’s for Apple displays only.