  Let $\nu$ be a self similar measure on $\mathbb{R}^d$, $d\geq 2$, and let
$\pi$ be an orthogonal projection onto a $k$-dimensional subspace. We formulate
a criterion on the action of the group generated by the orthogonal parts of the
IFS on $\pi$, and show that it ensures the dimension of $\pi \nu$ is preserved;
this significantly refines previous results by Hochman-Shmerkin (2012) and
Falconer-Jin (2014), and is sharp for projections to lines and hyperplanes. A
key ingredient in the proof is an application of a restricted projection
theorem of Gan-Guo-Wang (2024).
