The $n$-dimensional cross-polytope $P_n$ can be triangulated (divided into simplices, without adding vertices) in exactly $n$ different ways. Each of these has $2^{n-1}$ simplices.

A quick google search finds Example 6.3.14 in the book "Triangulations: Structures for Algorithms and Applications". They don't give a proof (and they have a typo, I think). This fact is also mentioned without proof in Example 4.9 of the paper "Splitting polytopes". Anyway, this seems to be a "standard fact".

Much less seems to be known about the triangulations of $Q_n$, the "rectified $n$-simplex".

Edit: This doesn't answer the question the poster is asking about decompositions.

The $n$-dimensional cross-polytope $P_n$ can be triangulated (divided into simplices, without adding vertices) in exactly $n$ different ways. Each of these has $2^{n-1}$ simplices.

A quick google search finds Example 6.3.14 in the book "Triangulations: Structures for Algorithms and Applications". They don't give a proof (and they have a typo, I think). This fact is also mentioned without proof in Example 4.9 of the paper "Splitting polytopes". Anyway, this seems to be a "standard fact".

Much less seems to be known about the triangulations of $Q_n$, the "rectified $n$-simplex".

The $n$-dimensional cross-polytope $P_n$ can be triangulated (divided into simplices, without adding vertices) in exactly $n$ different ways. Each of these has $2^{n-1}$ simplices.

A quick google search finds Example 6.3.14 in the book "Triangulations: Structures for Algorithms and Applications". They don't give a proof (and they have a typo, I think). This fact is also mentioned without proof in Example 4.9 of the paper "Splitting polytopes". Anyway, this seems to be a "standard fact".

The $n$-dimensional cross-polytope $P_n$ can be triangulated (divided into simplices, without adding vertices) in exactly $n$ different ways. Each of these has $2^{n-1}$ simplices.

A quick google search finds Example 6.3.14 in the book "Triangulations: Structures for Algorithms and Applications". They don't give a proof (and they have a typo, I think). This fact is also mentioned without proof in Example 4.9 of the paper "Splitting polytopes". Anyway, this seems to be a "standard fact".

Much less seems to be known about the triangulations of $Q_n$, the "rectified $n$-simplex".