Contents 1. Introduction 1 2. The rank 3 algebra F 3 3. Indefinite subalgebras from subroot systems 5 4. Indefinite subalgebras of F from the Weyl group 9 5. Borcherds algebras as subalgebras of hyperbolic KM algebras 12 6. Appendices 15 References 17We dedicate this work to the memory of our friend Peter Slodowy.
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