Some cubic modular identities of Ramanujan

There is a beautiful cubic analogue of Jacobi's fundamental theta function identity: θ⁴₃ = θ⁴₄ + θ⁴₂. It is ([formula cannot be replicated] q^n²+nm+m²)³ = ([formula cannot be replicated] ω^n-mq^n²+nm+m²)³ + ([formula cannot be replicated] q^{(n+1/3)²+(n+1/3)(m+1/3)+(m+1/3)²})³. Here ω = exp(2π i/3). In this note we provide an elementary proof of this identity and of a related identity due to Ramanujan. We also indicate how to discover and prove such identities symbolically