Identity for $4\sin^2\alpha\sin^2\beta\sin^2\gamma$ expressed as product of terms $(\pm\sin\alpha\pm\sin\beta\pm\sin\gamma)$

It can be done using Heron's formula (*) as stated in the comments, but notice that this is simply the constraint (or the Zariski closure of the constraint) on the sines of three angles to be sines of a triangle. I don't think there is a lower-degree polynomial expressing the same condition. This raises the question of whether a conceptual solution exists, avoiding the details of classical geometry.

[(*) Start from Heron's formula (squared), replace Area by $abc/4R$, replace $a,b,c$ by $2R\sin \alpha, 2R \sin \beta, 2R \sin \gamma$. Factors of $R$ will disappear from the final result, leaving the formula on sines.

The "well known" formula at the top of the question is a similar restatement of the fact that the area of a triangle is the sum of the small triangles formed by joining vertices to the center of the circumscribed circle.]