Given,
$x^{12}-x^7-x^6-x^5+1 = 0\tag1$
This has Lehmer’s decic polynomial as a factor,
$x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1=0\tag2$
hence one of its roots is the smallest known Salem number. All ten roots obey the beautiful cyclotomic relation,
$x^{630}-1=\frac{(x^{315}-1)(x^{210}-1)(x^{126}-1)^2(x^{90}-1)(x^{3}-1)^3(x^{2}-1)^5(x-1)^3} {(x^{35}-1)(x^{15}-1)^2(x^{14}-1)^2(x^{5}-1)^6\, x^{68}}$
found by D. Broadhurst. But this was back in 1999 (paper here). Has anything similar for other Salem numbers been found since then?