why is this figure not a simplicial complex?

Question

Is it because the intersecting free 1 simplex at bottom right. Would it be a simplicial complex if I removed that part?

Answer 1

Not just that. The fact that the triangle below is not given a proper structure of simplex must be taken into account too: keep in mind that any two simplexes must intersect in a face. So, it should be split in two triangles by means of an additional segment joining the midpoint of the side with three vertices with the opposite vertex.

Answer 2

Intersection of two simplex is not a simplex.