The elements of the solar system, (.) may be used to compute Cartesian coordinates and velocities of the planets. Here is how.

We use this formula for an ellipse:

r = A/(1 + e cos θ)
The origin is at a focus of this ellipse.
The above reference uses these terms:
a = semi-major axis
Half the long axis of the ellipse (a = A(1/(1+e) + 1/(1–e))/2 = A/(1–e2))
e = eccentricity
Departure from circle; see e in equation.
i = inclination
angle between plane of Earth’s orbit and plane of planet’s orbit.
Ω = longitude of the ascending node
Where the planet passes to the North side of the Earth’s orbital plane. Measured as an angle from a somewhat arbitrary point in the Earth’s orbit.
ϖ = longitude of perihelion
Where the planet is closest to the Sun. Measured from the same arbitrary point. This is the sum of two angles, the first in the Earth’s orbit, to the ascending node, and the rest in the planet’s orbit.
L = mean longitude
The angle which the planet would be at now, if it had been moving at its constant average angular velocity since perihelion. (That’s weird!) Measured from the same arbitrary point. L changes by 2π each orbit and is linear in time.
See too JPL’s Glossary.
Here is the plan, and code, to compute x, y, z:
Use e
Consider a unit ellipse with a focus at the origin, eccentricity = e, and X-axis as major axis. I.e. convert r = A/(1 + e cos θ) to Cartesian coordinates.
Use L
Locate body (x, y) on orbit iterating Kepler’s equal area equation.
Use a
Transform coordinates by scale factor A/(1–e2).
Use ϖ–Ω
Rotate ellipse.
Use i
Rotate in 3D about line of nodes by inclination.
Use Ω
Rotate line of nodes by longitude of the ascending node.
The last four steps are matrix multiplies, if this makes the code or the analysis easier.
Converting the above ellipse formula to rectangular coordinates we get: The length of the major axis is 2/(1–ε2).