these roots are real, then the subalgebra can be an indefinite KM algebra; however, we will also exhibit examples where the new simple roots are imaginary, and the resulting subalgebra is a Borcherds algebra [@B1; @Jur1; @Nie]. Two subalgebras are said to be $W$-equivalent if their respective simple roots are related by a Weyl transformation from the Weyl group of the original algebra. Geometrically, some of the non-$W$-equivalent subalgebras of indefinite KM algebras can be understood in terms of conical sections. Generally there are three ways of slicing indefinite (and hyperbolic) KM algebras corresponding to the intersections of the light-cone with a hyperplane in the dual of the Cartan subalgebra. That dual space is equipped with an indefinite bilinear form, $(\alpha,\beta)$, and depending on whether the hyperplane is space-like, light-like or time-like (meaning that its normal vector, $\nu$, satisfies $\nu^2 = (\nu,\nu) < 0$, $\nu^2 = 0$, or $\nu^2 > 0$) one obtains elliptic, parabolic or hyperbolic sections corresponding to a decomposition of the original KM algebra with respect to a finite, an affine or an indefinite subalgebra. (The latter need not be hyperbolic, even if the original algebra is hyperbolic.)

Previous attempts to understand indefinite KM algebras were almost 