I have a slightly broken dining chair made of pine.

Description

The back of the chair has two thin pieces of pine (about 6 or 7 mm
thick, 4-5 cm wide) crossing each other, and at the point of cross,
each piece has half a cut out so they cross flush. So if you imagine
you are looking at an X from the back, the bottom right part of the X
has broken about halfway past the intersection, i.e. about 3/4 along
the whole piece going from top left to bottom right.

The break half way along has zig zagged and is thinner toward the end
of the break. i.e. it is was not a clean break. It looks to have
snapped and been pealed off if you like. Hard to explain.

The Question

I would like to glue the two pieces back together. I'm thinking of no
nails. Is this the right type of glue to use?

Any more suggestions on how to complete this are also most welcome
please.

I guess once glued I will also have to lightly sand both sides to
prevent any splinters?

Thanks.