Short proof of the coincidence of left dom.dim., right dom.dim. and relative dom.dim. for semi-primary left QF-3 rings?

Question

is there a short and/or elementary proof of the following fact (which is taken from theorem 7.7 from H. Tachikawa, ‘‘Quasi-Frobenius Rings and Generalizations’’, Springer Lecture Notes in Mathematics):

Let $A$ be a semi-primary left QF-3 ring with minimal faithful left ideal $Ae$. Then the left dominant dimension of $A$ coincides with its dominant dimension relative to $Ae$. Moreover, the left dominant dimension coincides with the right dominant dimension.

Thanks for the help!