For gluing, I will squirt a small amount into the groove and spread it down in a bit with a thin stick. Next I'll smear some on the sides of the spline to fully cover and press it in.
The glue will be pushed around, so with the slot, it'll get pushed into the groove and the spline will have it pushed out to the corner peak.
I'll always clamp to push the spline into the slot by having a flat on the spline and a small 45 degree corner block for the inside of the box. I hate it when there is the glue-filled gap showing so this makes sure it is pressed in to the max.
Of course one can cross clamp in the spines from just the outside, but I find that that extra minute or more of gluing up two splines before clamping can let them seize up before being pressed in.