How to construct symmetric and positive definite

How to construct symmetric and positive definite $A,B,C$ such that $A+B+C=I$?

In an attempt to formulate a answer to this (in)famous question

How does one prove the matrix inequality $\det\left(6(A^3+B^3+C^3)+I_{n}\right)\ge 5^n\det(A^2+B^2+C^2)$?

I'm trying to construct three $n\times n$ matrices $A,B,C$ that are (a) symmetric, (b) positive definite, (c) add to $I_n$ . Note that I've already decided to restrict attention to the reals and I have replaced Hermitian by symmetric (which IMO is difficult enough).

My unsuccessful tries are a wild mixture of two extremes:

Make a random square matrix $H$ and form $A = H^TH$ . Make another random square matrix $H$ and form $B = H^TH$ . In the same way, form $C = H^TH$ . Then $A,B,C$ are symmetric and positive definite. But in general $A+B+C \ne I$ .
Generate random numbers for $A_{ij} = A_{ji}$ , $B_{ij} = B_{ji}$ and form $C_{ij} = C_{ji} = I_{ij}-A_{ij}-B_{ij}$ . Then $A,B,C$ are symmetric and $A+B+C = I$ , but it cannot be guaranteed that these are positive definite matrices.

So the question is: how can the three requirements (a) , (b) , (c) be fulfilled at the same time, while keeping $A,B,C$ yet as random as possible? My plan is to do numerical experiments and eventually find a counter example. I have all the ingredients to do it, except this.