The statistical geometry of dispersing Lagrangian clusters of four particles (tetrahedra) is studied by means of high-resolution direct numerical simulations of three-dimensional homogeneous isotropic turbulence. We give evidence of a self-similar regime of shape dynamics characterized by almost two-dimensional, strongly elongated geometries.

Bartlett's decomposition provides the distributional properties of the elements of the Cholesky factor of $A=G^TG$ where the elements of $G$ are i.i.d. standard Gaussian random variables. In this paper the most general case where the elements of $G$ have a joint multivariate Gaussian density is considered.