Looks pretty nice, and with the Lull name, is an antique. The springs under body give an improved ride over the solid construction of most sleighs.
This is known as the Portland sleigh body. Very common, with the other style known as an Albany with it's curved body. The Albany type is often called the Santa Claus sleigh.
In my experience, the small sleigh type shown is NEVER built with a cutunder. Sleighs on bobs (front and back short runners, where the front is steerable like a carriage), might have a raised front driver's seat, so it appears cutunder. These would be larger passenger sleighs, holding a rear area behind the driver's seat. Runners could be straight the length of the sleigh or on bobs as mentioned.
With longer straigh runners, the horse or horses, have to push sideways a bit for turns. Bobs are like wheels, turn shorter, a bit easier but require heavier construction to support the turning mechanism.
A cutunder on a sleigh would not serve the same purpose it does on a carriage, allowing wheels to turn shorter. Cutunder on sleigh would be more likely to be just decorative.
I would only use a sleigh if it was a cut under. I am too afraid of tipping.
OK. If you know nothing about sleighs, let me educate you:
They do not come with "cut unders" because...wait for it...they don't have wheels. Amazing, but true. They have steel ice "runners" - what one would call "blades" on ice skates. Sleighs slide, they don't roll. They slide to go forward, go backwards, to turn, to move. The whole body pivots as a whole - it goes wherever those fixed steel blades go.
99% of the sleighs made (both then and now) have runners extending from the curl in the front to a foot or more beyond the back seat, making for a very stable platform.
Those that don't have one front-to-back-set of runners tend to be the big multi-passenger sleighs. Most of these big sleighs are "bob sleighs" - they come equipped with two sets of short ('bobbed') runners - one fore and one aft. But they aren't cut under either.