I look at $ 1, i, j, k, l, m, n, o $ the standard base of the (complexed at will) octon ions ($ mathbb {O} $ for the octon ions).

To let $ a = x_1.1 + ldots + x_8.o $. $ b = x_9.1 + ldots + x_ {16} .o $ and $ c = x_ {17} .1+ ldots + x_ {24} .o $, Where $ x_1, ldots, x_ {24} $ are indefinite about the basic field (pron $ mathbb {C} $).

I denote by $ L_a $ the $ 8 times $ 8 Matrix that represents the left multiplication with $ a $ in the $ mathbb {O} simeq mathbb {C} ^ 8 $ and $ R_a $ the $ 8 times $ 8 Matrix that represents the correct multiplication with $ a $, Similar names for $ b $ and $ c $, I want to calculate the characteristic polynomial of the symmetric matrix:

$$ S = R_a L_b L_c + {} ^ {t} (R_a L_b L_c), $$

Where $ {} ^ {t} X $ is the transpose of $ X $,

I tried Macaulay2 and this calculation seems to go far beyond what my computer (which is supposed to be a fairly powerful portable workstation) offers.

A simple reformulation of the eigenvalue problem on a well-chosen basis (namely let $ mathbb {H} $ be the quaternionic subalgebra of $ b $ and $ c $, Splits $ mathbb {O} $ how $ mathbb {H} bigoplus mathbb {H} .e $, Where $ e $ is orthogonal to $ mathbb {H} $ and take a base adapted to this decomposition) shows that:

$$ (T – mathrm {Re} (( overline {b} c) overline {a})) ^ 4 textrm {divides} det (S-T.id), $$

Where $ mathrm {Re} (z) $ is the real part of $ z in mathbb {O} $,

I put $ f (T) = dfrac {det (S-Tid)} {(T – mathrm {Re} ((bc) overline {a}) ^ 4} $, A variety of calculations over finite fields and specialization of the $ x_i $ random values suggests that $ f (T) $ is indeed a square, we say $ f (T) = g (T) ^ 2 $, Where $ g $ is a quadratic polynomial in $ T $,

I would like a closed expression of $ g (T) $, May it be a clean formula $ a, b $ and $ c $ or a dirty "in coordinates" polynomial. I would be very happy about any suggestion. I would also be interested in a theoretical argument that shows that $ f (T) $ is indeed a square.

Many thanks!