Symmetric subalgebras of monocomposition algebras (Q1823301)

Let \(A\) be an NMU-algebra (in Russian: HME-algebra) that is a nonassociative algebra over a field \(k\), \(\mathrm{char}\;k\neq 2\), with a nondegenerate symmetric bilinear form \(N(x,y)\) such that \[ N(x^ 2,xy+yx)=2N(x,x)N(y,y)\quad \text{for all}\quad x,y\in A. \] Assume \(A\) admits \({\mathbb Z}_ 2\)-graduation \(A=A_ 0\oplus A_ 1\), where \(A_ 0\) is a nonisotropic subspace in \(A\) and \(A_ 1=A^{\perp}_ 0\). For any two finite-dimensional spaces \(B\) and \(C\) with fixed bases the author determines the multiplication table in \(A=B\oplus C\) which makes \(A\) an NMU-algebra, where \(B=A_ 0\) and \(C=A_ 1.\) Let \(A\) be a finite-dimensional commutative NMU-algebra, \(\mathrm{char}\;k\neq 2,3\), with a semisimple derivation \(d\) with eigenvalues \(0,\pm 1\). Then \(A=A(0)\oplus A(1)\oplus A(-1)\), where \(A(i)=(x\in A, d(x)=ix)\). Moreover, \(A(i)A(j)\subseteq A(i+j)\), \(A(\pm 1)^ 2=0\), \(A(1)\oplus A(- 1)=A(0)^{\perp}\), \(\dim A(-1)=\dim A(1)\) and \(A(1)\), \(A(-1)\) are completely isotropic. Some results of the paper concern commutative NMU-algebras with \(\dim A_ 1=1\). Finally, the author gives an example of an 11-dimensional commutative algebra \(A\) which has no \({\mathbb Z}_ 2\)-graduation and \(\Aut A\simeq k^ 4\).