The real Albert algebra is a Jordan algebra

Conjugation of the $(i,j)$ entry of $XY$ gives $\overline{(XY{)}_{ji}}={\sum }_r\overline{X_{jr}{Y}_{ri}}={\sum }_r\overline{Y_{ri}}\overline{X_{jr}}={\sum }_r{Y}_{ir}{X}_{rj}=(YX{)}_{ij}\mathrm{.}$ Thus $XY+YX$ is Hermitian off the diagonal. Its $i$th diagonal entry is ${\sum }_r(X_{ir}\overline{Y_{ir}}+Y_{ir}\overline{X_{ir}})\mathrm{.}$ Each summand is fixed by conjugation, since its conjugate is the same sum in reverse order; by [F2] the fixed elements are real. Therefore $m(X,Y)$ is in $H_3(\mathbb{O})$. For bilinearity, [F5] shows that quaternion multiplication is real-bilinear and quaternion conjugation is real-linear. The pair formula in Real octonions by Cayley–Dickson doubling therefore makes octonion multiplication real-bilinear. Each entry of the matrix product in [F1] is a sum of three octonion products, so matrix multiplication is real-bilinear; consequently $D(X,Y)=XY+YX$ and $m(X,Y)=D(X,Y)\mathrm{/}2$ are bilinear. The formula for $m$ makes it commutative. The quaternion coordinate formula gives $\widehat{1}a=a\widehat{1}=a$ and $\bar{0}=0$, so the pair formula gives $(1,0)(a,b)=(\widehat{1}a-\bar{b}0,b\widehat{1}+0\bar{a})=(a,b)$ and $(a,b)(1,0)=(a\widehat{1}-\bar{0}b,0a+b\bar{\widehat{1}})=(a,b)$; hence $1_{\mathbb{O}}=(1,0)$ is a two-sided unit. The identity matrix is in $H_3(\mathbb{O})$ and is a two-sided unit for the matrix product, hence $m(I,X)=X$.

Let $1,i,j,k,\mathrm{\ell },i\mathrm{\ell },j\mathrm{\ell },k\mathrm{\ell }$ be the ordered real basis of $\mathbb{O}=\mathbb{H}\times \mathbb{H}$ specified by the pair formula. The three diagonal matrix units, together with the eight Hermitian matrices having one of these octonions in the $(1,2)$, $(2,3)$, or $(3,1)$ slot and its conjugate in the transposed slot, form a real basis of $H_3(\mathbb{O})$: the three real diagonal coordinates and the three independent octonion entries are recovered uniquely from any Hermitian matrix. There are $3+3\cdot 8=27$ basis elements.

Set $D(X,Y):=XY+YX$. With $m=D\mathrm{/}2$, bilinearity gives $8(m(X,m(Y,m(X,X)))-m(m(X,Y),m(X,X)))=D(X,D(Y,D(X,X)))-D(D(X,Y),D(X,X))\mathrm{.}$ The right side is a polynomial homogeneous of degree three in $X$ and degree one in $Y$, with coefficients in the real basis of $H_3(\mathbb{O})$ fixed here.

Write $X={\sum }_i{x}_i{E}_i$ and $Y={\sum }_{\mathrm{\ell }}{y}_{\mathrm{\ell }}{E}_{\mathrm{\ell }}$ in this basis. The coefficient of $x_i{x}_j{x}_k{y}_{\mathrm{\ell }}$ in the degree-$(3,1)$ expression in step 2.1 is the sum over the distinct permutations $(p,q,r)$ of $(i,j,k)$ of $D(E_r,D(E_{\mathrm{\ell }},D(E_p,E_q)))-D(D(E_r,E_{\mathrm{\ell }}),D(E_p,E_q))\mathrm{.}$ This follows by distributing each bilinear $D$; when indices repeat, distinct permutations count the distinct placements in the cubic monomial exactly once. The script in [F4] computes all ${27}^2=729$ ordered basis-pair products to encode $D$, rejecting any non-Hermitian result, then checks these coefficients for every sorted triple $0\le i\le j\le k<27$ and every $0\le \mathrm{\ell }<27$. It checks $\left(\genfrac{}{}{0px}{}{29}{3}\right)\cdot 27=\mathrm{98,658}$ monomial coefficients, or $\mathrm{98,658}\cdot 27=\mathrm{2,663,766}$ output coordinates. Its signed integer arithmetic returns zero in every coordinate, with the convention witnesses $ij=k$, $(ij)\mathrm{\ell }=k\mathrm{\ell }$, and $i(j\mathrm{\ell })=-k\mathrm{\ell }$ all true. Since the basis coordinates of the polynomial are exactly the coefficients enumerated, the polynomial vanishes for every $X,Y$.

step 1.1 and step 3.1 establish closure, commutativity, bilinearity, a unit, and the Jordan identity in [F3]. Hence $H_3(\mathbb{O})$ is a unital real Jordan algebra.