More characterisations are in the book of Haim Brezis (Analyse fonctionnelle), at the appendix of Chapter 5. I will copy two of these below, toghether with the references:

1. If $ \dim(E)\geq 2 $ and every subspace $ X\subset E $ of dimension $ 2 $ is the image of a bounded projector $ P $ such that $ \|P\| = 1 $, then $ E $ is isometric to a Hilbert space (Kakutani, Japaneise Journal of Mathematics, 1939);
2. if $ \dim(E)\geq 3 $ and the map $ T $, defined as the identity on the unit ball and as $ u/\|u\| $ when $ \|u\|\geq 1 $, is lipschitzian with constant $ 1 $, then $ E $ is isometric to a Hilbert space (de Figueiredo; Karlovitz, Bulletin of the American Mathematical Society, 1967).

Also, if $ E $ is isomorphic to all its infinite-dimensional subspaces, then it is isomorphic to a separable Hilbert space (Gowers, Annals of Mathematics, 2002).

More characterisations are in the book of Haim Brezis (Analyse fonctionnelle), at the appendix of Chapter 5. I will copy two of these below, toghether with the references:

1. If $ \dim(E)\geq 2 $ and every subspace $ X\subset E $ of dimension $ 2 $ is the image of a bounded projector $ P $ such that $ \|P\| = 1 $, then $ E $ is isometric to a Hilbert space (Kakutani, Japanese Journal of Mathematics, 1939);
2. if $ \dim(E)\geq 3 $ and the map $ T $, defined as the identity on the unit ball and as $ u/\|u\| $ when $ \|u\|\geq 1 $, is lipschitzian with constant $ 1 $, then $ E $ is isometric to a Hilbert space (de Figueiredo; Karlovitz, Bulletin of the American Mathematical Society, 1967).

Also, if $ E $ is isomorphic to all its infinite-dimensional subspaces, then it is isomorphic to a separable Hilbert space (Gowers, Annals of Mathematics, 2002).